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Basis sets featuring single-exponent radial functions for each of the n subshells and orthogonality of the radial parts for dierent values of n within the same have been generated for elements 1 to 54 of the Periodic Table, by minimizing the total energy for dierent spectroscopic states. The derived basis sets can be fairly dubbed as MAP (minimal atomic parameter / Moscow-Aachen-Paris) basis sets. We show that fundamental properties (total energy, radial expectation values, node positions, etc.) of the generated MAP orbital sets are astonishingly close to those obtained with much larger basis sets known in the literature, without numerical inconsistencies. The obtained exponents follow simple relations with respect to the nuclear charge Z. Possible further applications, trends, and limitations are discussed.

We present the implementation of a variational finite element solver in the HelFEM program for benchmark calculations on diatomic systems. A basis set of the form $\chi_{nlm}(\mu,\nu,\phi)=B_{n}(\mu)Y_{l}^{m}(\nu,\phi)$ is used, where $(\mu,\nu,\phi)$ are transformed prolate spheroidal coordinates, $B_{n}(\mu)$ are finite element shape functions, and $Y_{l}^{m}$ are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree--Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin-density approximation (LDA), generalized gradient approximation (GGA) and the meta-GGA level can be used through an interface with the Libxc library; meta-GGA and hybrid DFs aren't available in other fully numerical diatomic program packages. Finite electric fields are also supported in HelFEM, enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HelFEM and the novel basis set procedure are demonstrated by reproducing the restricted open-shell HF limit energies of 68 diatomic molecules from the $1^{\text{st}}$ to the $4^{\text{th}}$ period with excellent agreement with literature values, despite requiring \emph{orders of magnitude} fewer parameters for the wave function. Then, the electric properties of the BH and N2 molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed in the literature references. Finally, HF, LDA, GGA, and meta-GGA calculations of the atomization energy of N2 are performed, demonstrating the superb accuracy of the present approach., Comment: 33 pages, 2 figures. Minor changes to text

Actinide compounds are very intriguing objects for the quantum chemistry because, on the one hand, these compounds are of great scientific and technological interest and, on the other hand, quantitative first principle based modeling of their electronic structure is extremely difficult because of strong relativistic effects and complicated electron correlation pattern. The efficiency of high-level all-electron relativistic methods in applications to complex actinide systems of practical interest is questionable and more economical but sufficiently accurate approaches to the studies of such systems are preferable. Recently, generalized relativistic effective core potentials (GRECPs) have been generated for actinides to perform accurate calculations of electronic structure and properties of their compounds with moderate computational cost. The accuracy of different GRECP versions is analyzed in atomic calculations and their applications to molecular and cluster calculations are reviewed. The results are compared with available experimental data and other theoretical studies. © 2015 Wiley Periodicals, Inc.

An open-shell Hartree–Fock (HF) theory for spin-dependent, two-component relativistic calculations, termed the Kramers-unrestricted HF (KUHF) method, is developed. The present KUHF method, which is formulated as a relativistic counterpart of nonrelativistic UHF, is based on quaternion algebra and partly uses time-reversal symmetry. The fundamental characteristics of KUHF are discussed in this study. From numerical assessments, it was revealed that KUHF gives a corresponding solution to nonrelativistic UHF; furthermore, KUHF properly describes spin-orbit interactions. In addition, KUHF can improve the self-consistent field convergence behavior in spin-dependent calculations, for example, for f-block elements.

In this work, the multireference Hartree–Fock configuration interaction method (MRHFCI) was applied to calculate the permanent electrical quadrupole moment of the hydrogen fluoride molecule, using the atomic bases STO-6G and CFG-6G. We have obtained MRHFCI functions, with energies comparable to the full CI ones, providing dipole and quadrupole moments in close agreement with the experimental values and with reduced MRHF basis, that is, with significant lower numbers of configurations. © 2012 Wiley Periodicals, Inc.

Comparative quantitative-structure-toxicity-relationship models developed using various quantum chemical descriptors calculated at semiempirical (AM1, PM3), Hartree–Fock and density functional theory (DFT) (B3LYP) methods, are compared for predicting the toxicity (pIGC50) of 69 aliphatic compounds toward Vibrio fischeri. The multiple linear regression equations, which can assist in modeling toxicity of these compounds, are developed using molecular descriptor, log P (1-octanol/water partition coefficient) in conjunction with two quantum chemical descriptors, electrophilicity index (ω) and energy of the lowest unoccupied molecular orbital (ELUMO). The calculated toxicity values (PIGC50) based on the model equations are compared with the experimental data. In general, the stepwise model regression equations based on quantum chemical descriptor ω in combination with molecular descriptor log P, calculated at DFT(B3LYP) with 6-311G (d,p) and DGDVZP basis sets show better agreement with the experimental data. The electronic origin of the toxicity dependence on descriptors like log P, ω and ELUMO is traced to the cumulative contribution of the lipophilicity and electrophilicity of the chemical moieties to their toxic behavior. © 2012 Wiley Periodicals, Inc.

The Hartree–Fock (HF) method, widely used for the calculation of ground state properties, in recent years has been extended for calculating excited states. In this article, we develop an approach, which does not require time consuming calculations and can give a good approximation for the excitation energies. The method is based on the fact that the subspaces of the occupied and virtual orbitals are mutually orthogonal. We test the accuracy of our method by evaluating the excited state wavefunctions and the corresponding energies for the Li, N, F, Ne, and Na atoms and for the BeH and CH molecules, and we found good agreement with configuration interaction and experimental results. © 2012 Wiley Periodicals, Inc.

We show that, in the high-density limit, restricted Moller-Plesset (RMP) perturbation theory yields E (2) RMP = π

The title compound, N′-benzylidene-N-[4-(3-methyl-3-phenyl-cyclobutyl)-thiazol-2-yl]-chloro-acetic acid hydrazide, has been synthesized and characterized by elemental analysis, IR, 1H and 13C NMR, and X-ray single crystal diffraction. The compound crystallizes in the orthorhombic space group P 21 21 21 with a = 5.8671 (3) A, b = 17.7182 (9) A, and c = 20.6373 (8) A. Moreover, the molecular geometry from X-ray experiment, the molecular geometry, vibrational frequencies, and gauge-including atomic orbital 1H and 13C chemical shift values of the title compound in the ground state have been calculated by using the Hartree–Fock and density functional methods (B3LYP) with 6-31G(d) and 6-31G(d,p) basis sets. The results of the optimized molecular structure are exhibited and compared with the experimental X-ray diffraction. Besides, molecular electrostatic potential, Frontier molecular orbitals, and thermodynamic properties of the title compound were determined at B3LYP/6-31G(d) levels of theory. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

We report a new mathematical result: it is possible to construct a spectral representation of the two particles Coulomb potential in the form of |r − r′|−1 = ∑λλg(r) gλ(r′). We call this formula λ-decomposition. Two special nontrivial cases of λ-decomposition are reported together with the numerical analysis of the convergence for one of them. It is shown how λ-decomposition allows to construct a new fast algorithm for Hartree-Fock exchange operator calculation, in which the calculation of electron repulsion integrals (ERIs) is completely avoided. The connection between the new method and the resolution of identity and Cholesky decomposition based approaches has been established. Finally, the accuracy of ERIs evaluation within the new approach has been studied numerically. The results demonstrate that it is possible to achieve the accuracy of 10−10 for the ERIs in wide range of their orbital exponents with relatively small number of terms in λ-decomposition. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

There are well-established mathematical relationships for the energy of a diatomic molecule (the exact energy, but also both the Hartree-Fock and Thomas-Fermi energies) in the limit of large nuclear charge of the atoms and small internuclear distances. We present calculated energies for a number of homonuclear diatomic molecules at short internuclear distances for numerical comparison with these established relationships. Intriguingly, they hold in practice for relatively small nuclear charges and for relatively long internuclear distances.

The quantum chemistry of finite aperiodic graphene flakes is a matter of considerable interest because of the anticipated technological importance of such objects. Since real aperiodic graphene flakes will in general be composed of many thousands of carbon atoms, theoretical methods appropriate to such large molecules would need to be used for the ab initio quantum calculation of their properties. The Kernel energy method is discussed here, and it is shown to be accurately applicable to graphenes and analogous extended aromatic molecules. It is necessary to define the kernels of a graphene molecule in a new way because of the extensive aromaticity, which defines its electronic structure. The kernels used in the reconstruction of the full graphene sheet preserve the total number of π-electrons, Clar sextets, and the approximate overall aromaticity. Sivaramakrishnan et al. [J Phys Chem A, 2005, 109, 1621] define similar “ring conserved isodesmic reactions (RCIR).” The principal innovation of this article is the suggestion that kernels may be mathematically extracted from an extended aromatic molecule such as graphene by a fissioning of aromatic bonds. Hartree Fock (HF) and Moller-Plesset (MP2) chemical models using a Gaussian basis of 3-21G orbitals are used to calculate the total energy of a graphene flake. This demonstration calculation is performed on a graphene flake in which dangling bonds are saturated with hydrogens (C78H26) composed of a total of 104 atoms arranged in 27 benzenoid rings. The KEM with both types of chemical model are shown to be accurate to nearly 1 kcal/mol, of a total energy, which is nearly 3000 atomic units, that is, with an absolute error within “chemical accuracy” and a relative error of the order of 5 × 105% of the total energy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

In this article, we performed spin-unrestricted Hartree Fock (UHF) based coupled-cluster (CC) and Bruckner CC double calculations of methylene, fluorine, HHeH diradical, and ozone. Mukherjee-type (Mk) Multireference coupled-cluster (MRCC) calculations of these species were also conducted by the use of UHF natural orbital (UNO) and UHF localized natural orbitals (ULO) as reference orbitals to elucidate size-consistent error. Comparing calculated results of spin-unrestricted Hartree-Fock based coupled-cluster (UCC) and spin-unrestricted Bruckner-Double (UBD) with those of MkMRCC, it was found that spin-contamination errors of UCC and UBD solutions still remained, though this type of errors for these solutions was considered to be negligible in general. The comparison also revealed scope and applicability of approximate spin-projection (AP) scheme for UCC and UBD, indicating that the AP procedure corrected spin-contamination errors to yield good agreement with MRCC in energy. We also performed pure- and hybrid- density functional theory (DFT) calculations of the same systems, and those calculations provided reasonable singlet-triplet energy gaps, which were qualitatively good agreement with those of MRCC, AP-UCC, and AP-UBD calculations. Implications of the computational results are discussed in relation to applicability of hybrid DFT to large superexchange coupled systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Total statistical energy differences for electronic excitation energies and ionization potentials are expanded in a McLaurin series leading to one-electron energy terms plus correction terms. The different nature of the correction to the one-electron values for the above properties in the Hartree–Fock–Slater scheme as compared to the Hartree–Fock method is commented on. Examples are provided comparing excitation energies and ionization potentials from experiment, transition state method, total statistical energy differences, and one-electron estimates for several sulfur nitrides.

Excitation energies and oscillator strengths in the multiconfigurational time-dependent Hartree-Fock (MCTDHF) approximation are presented for the O2 molecule. Variations of excitation energies with respect to the diffuse character of the basis set are examined. The equivalence between oscillator strengths calculated in the dipole length and dipole velocity approximation in the MCTDHF approximation is investigated.

A technique based on the multipole expansion is used to calculate accurately and for a negligible computing effort the long-range contributions in helical systems. The only quantities to be evaluated are the multipole moments of the charge distribution in the reduced unit cell. The method is applied to the 2*3/1 isotactic polypropylene helix.

A generalization of Hartree theory derived recently provides rigorous upper bounds for the Hartree-Fock energies of electronic systems. Augmentation of this theory by a practical lower bound for correlation-exchange energies yields a generalization of Hartree-Slater theory which provides lower bounds for Hartree-Fock energies.

Encouraged by the results for one electron systems, using a properly balanced Gaussian basis, the Hartree–Fock–Dirac–Roothaan SCF equations are solved. All calculations are done with the relativistic molecular program package MOLFDIR, exploiting molecular double group symmetry to reduce integral storage requirements and to obtain a Fock-matrix for each symmetry representation. Closed shell SCF results are given for various atomic and molecular systems.

Projected Hartree--Fock calculations involving both spin and orbital angular momentum projection are reported for the lowest $sup 2$P state of Li. The value 0.0679 a.u. is obtained for the spin-dipolar hyperfine structure term, in very good agreement with the experimental result of 0.0690 a.u. The energy is -7.36516 a.u. compared to the Hartree--Fock result of -7.36507 and the experimental energy of -7.41016 a.u. (auth)

The time-dependent Hartree-Fock-Slater equation was used to derive an expression for the wave-vector- and frequency-dependent dielectric function ϵ(q, ω). The effect of including the Xα exchange in the derivation is to subtract a term from the Coulomb contribution to the dielectric function in the random phase approximation. Self-consistent LCAO Hartree-Fock-Slater energy bands were used to compute (q, ω) at several points q along the [100], [110], and [111] axes in the first Brillouin zone. The real and imaginary parts of the dielectric function were computed independently. It was found that for wave vectors in the first Brillouin zone, there is little difference between the random phase and Hartree-Fock-Slater dielectric functions.

A transition operator, involving one calculation per excitation process, is reviewed. Applications to photoelectron and Auger Spectroscopy are discussed. Relaxation is included. (auth)

A reformulation of the CIBS theory of Hartree-Fock energies is carried out in which the explicit dependence on virtual Orbitals is made manifest. It is found that the required CIBS correction functions are orthogonal to the virtual orbital space and are indirectly coupled to them. Consequently, an approximation that neglects virtual orbitals is presented. In addition a decoupled and an uncoupled approximation are suggested. For programming purposes an explicit set of linear equations suitable for interfacing with standard molecular orbital programs is derived.

The purpose of this paper is to indicate the importance of using relativistic self-consistent-field wave functions for the calculation of crystal-field radial integrals. In relativistic calculations the wave functions corresponding to j = l ± 1/2 are different and give different radial integrals. In order to facilitate comparison with nonrelativistic results it is convenient to apply the effective-operator technique in crystal-field calculations as has been done for a long time in hyperfine-structure analysis. In this paper, general formulas are given for the relativistic effective-field Hamiltonian with the corresponding radial integrals. Theoretical values of radial integrals calculated with both relativistic Hartree-Fock and Hartree-Fock-Slater type wave functions are given for Gd3+ and Er3+

Magnetic hyperfine structure constants (aa) have been calculated for all the atoms (in the ground states) up to radon (Z = 86) using the numerical Hartree–Fock wave functions of Froese. Agreement between the experimental and calculated values of aa is extremely good, indicating that HF wave functions form an adequate basis for calculating aa.

The LCAO Hartree-Fock-Slater method is evaluated by comparing results of this method obtained in an accurate STO basis for a series of small diatomic molecules with experimental data and with results of Hartree-Fock calculations. It is found that the Xα potential, if applied with the same care as the Hartree-Fock potential (i.e., comparable basis set quality, etc.), leads to results that are at least as close to experiment as Hartree-Fock results. Because the HFS method is relatively fast, it can be used easily for calculations on transition-metal complexes, even in calculations with STO basis sets of double-zeta quality or better. Calculated ionization energies, core-electron energy shifts, spectroscopic term energies, infrared and Raman data, and electronic structures calculated for transition-metal complexes compare excellently with experiment.

Absolute multiplet strengths for dipole allowed transitions between some excited quartet states of Ne+ isoelectronic sequence have been calculated in the Hartree-Fock approximation and Russel-Saunder's coupling scheme with the help of interchange theorem within the context of z expansion method and perturbation theory. Comparison of radial transition integrals of the present calculation with those computed numerically for the case of Ne+ shows that the difference is only 14 percent. As the accuracy of the method increases with increasing z, the results are likely to be more reliable for higher members of the isoelectronic sequence.

Predictions of the local density formalism approach to cohesive properties of covalently bonded solids (diamond and cubic BN) and prototype ionic system (LiF) are described using results of recently developed fully self-consistent numerical basis set LCAO-DVM approach. Comparisons with restricted Hartree-Fock results and experiments for cohesive energies and equilibrium lattice constants are presented. Some of the principal bonding mechanisms in these crystals are discussed in terms of the contributions of local exchange and correlation to the binding and the charge redistribution relative to the noninteracting atoms.

A numerical multiconfiguration Hartree-Fock (MCHF) approach to atomic structure calculations is described.

Fermi contact interaction constants (aa) have been evaluated for a large number of atoms using the numerical Hartree–Fock wave functions of Froese [1]. Agreement between the calculated and the experimental values of aa is rather poor, indicating Hartree–Fock functions cannot be used to describe accurately the electronic spin density at the nucleus.

A relationship between the effective principal quantum numbers of Hartree–Fock orbitals and the domains of integral transform orbitals, with homogeneous weight functions, is defined. This relationship is used to determine the nonlinear parameters of Hulthen type 0s and 1p orbitals for the ground states of the second and third rows of atoms and to explain the failure of the Hulthen 2d function as an adequate representation of a 3d Hartree–Fock orbital.

In the Hartree—Fock equations, as simplified by Slater, each electron is assumed to be acted upon by an exchange potential. The exchange potential, for an electron of a given spin, can be expressed in terms of the radius of the exchange hole surrounding the electron in question. This paper reports a simple calculation on the position dependence of the exchange hole radius in the Ne and Ar atoms, where there are equal numbers of spin-up and spin-down electrons, and in the Na atom, where the numbers of the spin-up and spin-down electrons differ by 1.

The complete spin-orbital formulation of the analytical first and second derivatives of the Hartree-Fock (HF) energy as well as the analytical first derivative of the correlated second-order Moller-Plesset perturbation energy (MP2) is presented. Some features of an efficient computational method to calculate these derivatives are described. The methods are applied to calculate the harmonic vibrational frequencies of ethylene, and the results are compared with experiment.

Calculations with the ppp model for alternant hydrocarbons of polyenic character, which exhibit the convergence difficulties in the usual Hartree-Fock iterative procedure are carried out. The standard process ends in oscillations of atomic charges when the geometry deviates enough from the prediction from a single Kekule structure. Deviations in geometry sufficient for non-convergence can be very small.

Electric field gradients (q) and nuclear quadrupole coupling constants for the neutral atoms with pp and dd configurations from helium (Z = 2) to radon (Z = 86) have been calculated using the numerical Hartree–Fock wave functions of Froese [1]. Comparison with the experimental values of eqQ is made wherever these values are available; the calculated and experimental values are found to be in excellent agreement.

From the examples of two hydrogen molecules and two neon atoms we have tried to show some of the characteristics of the dispersion term in an ab initio perturbation procedure; for example, the importance of different partitionings of the molecular Hamiltonian, the need for diffuse polarization functions, the rather good agreement between the Hartree-Fock dispersion term and the semiempirical results using the multipole expansion, and the importance of certain molecular orbitals.

By making the assumption that the addition (subtraction) of an electron from a many-body wave function may be represented by a single spin orbital, variationally determined, two distinct one-particle potentials are derived. They represent, respectively, the effective interaction of the (N - 1)-electron system with the Nth electron, and the effective interaction of the N-electron system with the (N + 1)th electron. Nonlocal exchange arises because of the indistinguishability of electrons. The “core” electrons are allowed to be correlated with each other and to be polarized by the electron in question. Eigenvalues to these potentials represent, respectively, ionization potentials and electron affinities, and the eigenfunctions are variationally determined spin orbitals (“standing-wave solutions”), termed the “natural transition orbitals.” The two-body density matrix of a reference state approximating an eigenfunction of the many-body Hamiltonian is required as input to the theory. Given this input, the method is linear, requiring no interations. In the limit of no correlation in the reference N-body state (provided the occupied orbitals span the Hartree-Fock space), the two potentials reduce to the unrestricted Hartree-Fock potential for occupied and virtual orbitals.

The self-consistent-field equations for the ground state of the argon atom have been solved for a range of values of the parameter C in the local approximation: to the Hartree-Fock exchange potential. The orbital solutions with and without the Latter tail modification are used to investigate the relative merits of the Slater (C = 1) and Kohn-Sham (C = 2/3) local exchange potentials. To do this various expectation values are calculated. Total and one-electron binding energies are calculated as the expectation value of the Slater determinant built up from the orbitals and by using the statistical approximation to that expression. It is this statistical approximation for the total energy which Kohn-Sham varied to obtain their local exchange potential. When computing one-electron binding energies for frozen orbitals in the statistical approximation, it is only in the Slater case (C = 1) that the eigen-values of the self-consistent-field equations may be so interpreted. The role of the virial theorem and the improvements yielded by scaling are discussed. The Kohn-Sham model compares better with Hartree-Fock than the Slater model with regard to total energy, one-electron binding energies and the virial theorem. Other choices for C, according to various criteria, are discussed with the view towards a better statistical approximation to the Hartree-Fock exchange potential.