Restricted Hartree-Fock Theory Research Articles

Decomposition of the Electronic Energy in Terms of Density, Density Coherence, and the Connected Part of the Two-Body Reduced Density Matrix.

We analyzed static and dynamic electron correlation by decomposing the total electronic energy of calculations by restricted Hartree-Fock theory, complete active-space self-consistent field (CASSCF) theory, and multireference configuration interaction (MRCI). We used three different schemes to break down the relative energy contributions to the potential energy curves for the dissociation of H2, F2, and N2. The first decomposition scheme involves the classical and nonclassical components of the energy. The second and third recognize the part of the energy that is not expressible in terms of the one-body reduced density matrix; this is called the connected energy. The unconnected component is further decomposed into a part calculable from the density and the part calculable from the density coherence. The first decomposition scheme shows that the sum of the one-electron energy and the classical two-electron energy contains a negligible portion of the static correlation. This quantity has a relatively small variance between the three levels, especially for CASSCF and MRCI. This provides an explanation of why multiconfiguration pair-density functional theory and multiconfiguration density-coherence functional theory are able to improve the CASSCF energy. The latter two decompositions show that the connected energy contains a significant portion of static correlation. The energy representable by either the density or the density coherence is significantly different at the three levels. Mixing the density and density coherence between different methods may lead to a systematic error in the bond dissociation energy and the equilibrium bond distance, indicating that the density energy component and the density coherence energy component both include a significant amount of both static and dynamic correlation. These wave function decompositions can be useful for developing new functionals for density functional theory, density-coherence functional theory, density matrix functional theory, and pair-density functional theory and for guiding expectations for these theories.

The inhomogeneous electron liquid in the ligand oestradiol compared and contrasted with that in oestrone

Following earlier theoretical work on a candidate hormone bonded with the Ru complex NAMI-A, here, we begin an extension to involve the ligand oestradiol. In particular, we shall compare and contrast the ground-state inhomogeneous electron liquid in oestradiol with that in the closely related hormone; oestrone. Both these systems are studied quantum-chemically at the level of restricted Hartree–Fock theory (RHF). In each case, the equilibrium geometries are recorded in tabular form. The sum of the RHF-occupied eigenvalues is related to the ground-state energy of the electron liquid in each case. Finally, a brief qualitative discussion is added of the way the ligand oestradiol can cause dimerisation of oestrogen receptors and its likely interest for cell biology and for human breast cancer.

<i>Ab Initio</i> Calculations for the Effect of Pressure on the Structural Properties of Si Nanoclusters

Lattice constant, bulk modulus, Young modulus, valence band width, conduction band width, energy gap, vibrational energy, and plasmon energy have been calculated under compression and tensile stresses in the range (0 ± 10 GPa) for 8, 54, 128 atom clusters of silicon by means of density functional theory method with restricted Hartree-Fock theory within the framework of large unit cell approach. It is found that the results deduced from eight atom cluster are in good agreement with the corresponding experimental values. On the other hand, bulk modulus, young modulus, valence band width, energy gap, and Plasmon energy increase (decrease) under compression (tensile), while volume decreases (increases) with compression (tensile). The vibrational energy has a minimum value at the ground state point. The conduction band width has no systematic behavior with pressure.

Spin–spin coupling constants and triplet instabilities in Kohn–Sham theory

Indirect nuclear spin–spin coupling constants calculated using restricted Hartree–Fock theory are unreliable since the usually dominant Fermi-contact (FC) contribution arises from a triplet perturbation of the electronic system, poorly described in the Hartree–Fock theory – in particular, at geometries close to the onset of triplet instabilities. These problems are usually but not invariably overcome in Kohn–Sham theory, which typically provides good spin–spin coupling constants. We here examine the sensitivity of spin–spin coupling constants to triplet instabilities in Kohn–Sham and Hartree–Fock theories by correlating the quality of the spin–spin coupling constants and the quality of the lowest triplet excitation energy for a number of small molecules. In general, the FC contributions are most stable in the local density approximation (LDA) and slightly less stable in the generalised gradient approximation (GGA); on the other hand, GGA coupling constants are usually more accurate than LDA constants. Importantly, although hybrid theory often gives better results than the GGA theory, it is also more susceptible to triplet instabilities (inheriting some of the problems of the Hartree–Fock theory) and therefore less reliable than the GGA theory for spin–spin coupling constants. For calculations of spin–spin coupling constants, we recommend the Perdew–Burke–Ernzerhof GGA exchange-correlation functional, which provides a good compromise of accuracy and robustness.

Enantioanalysis of R-deprenyl based on its molecular interaction with C70 fullerenes

Two enantioselective, potentiometric membrane electrodes based on [5,6]fullerene-C(70) (1) and diethyl (1,2-methanofullerene C(70))-71-71-dicarboxylate (2) immobilized in carbon paste, were designed for the enantioanalysis of R-deprenyl. The electrodes exhibited near-Nernstian slopes: 57.90 (1) and 59.00mV/decade of concentration (2), respectively with low limits of detection 5.9x10(-11) (1) and 9.6x10(-11)mol/L (2), respectively. The linear concentration ranges are between 10(-10) and 10(-4)mol/L (1) and between 10(-9) and 10(-4)mol/L (2), respectively. The different characteristics involved in the molecular interaction between R-deprenyl and C(70) fullerenes were explained, namely (i) the stability of each molecule and (ii) the explanation of the molecular mechanism of interaction, using restricted Hartree-Fock theory, 3-21G(*) RHF-basis set. Furthermore, two intermolecular forces of interactions confer the stability of the electrodes; electrostatic interaction and moderate hydrogen bond interaction. Stability and feasibility of all the generated structures involved in this analysis were supported by their respective fundamental frequencies and energy minima. R-deprenyl can be recovered with average recoveries higher than 99.10% (RSD<0.03%) from synthetic mixtures between R- and S-deprenyl. The high selectivity and enantioselectivity made possible the enantioanalysis of R-deprenyl in its pharmaceutical formulations.

Quantum-chemical study of the potential anti-cancer drug Ru–NAMI-A in complex with estrogen and the simulated VA and VCD spectra of estrogen, the Ru–NAMI-A drug, and possible/proposed estrogen–Ru–NAMI-A complexes

Following up on an earlier theoretical report by Knapp-Mohammady (Phys Lett A 372:1881–1884, 2008) on the ground state of the neutral Ru complex NAMI-A (trans-imidazoledimethylsulfoxide-tetrachlororuthenate), we first report here a quantum-chemical study of the effect of both oxidation and reduction of the parent molecule to form the anionic and cationic species. The new structures are compared with the equilibrium nuclear structure reported earlier for the neutral complex. We anticipate that one such Ru cluster, with potential as an anti-cancer drug, will interact via an appropriate receptor, rather than directly with DNA. A receptor for NAMI-A binding in here proposed to be the steroid hormone, estrogen, C18H24O2. The biomolecular structure of the dicomplex is predicted from restricted Hartree–Fock theory and density functional theory (DFT) calculations. The vibrational frequencies of NAMI-A and the dicomplex with estrogen are also reported. Some maps of the ground-state electron-density for the three neutral biomolecular species are finally presented. The use of vibrational spectroscopy, vibrational absorption (VA) and vibrational circular dichorism (VCD) are advocated to be measured, simulated and be used to understand the nature of the interaction of the Ru complex NAMI-A in complex with estrogen. Our aim in presenting these spectral simulations is to motivate the measurement of the VA and VCD spectra of estrogen, the Ru complex NAMI-A and finally of the estrogen–Ru NAMI-A complex. It should also be instructive to measure the VA and VCD spectra of estrogen and the estrogen receptor, both alone, together and finally together in the presence of the Ru NAMI-A complex to substantiate our claim that the Ru complex NAMI-A ties up estrogen, and hence prevents estrogen binding to the estrogen receptor.

Additive and non-additive information channels in orbital communication theory of the chemical bond

The molecular communication system in the resolution of the basis functions χ = {χ X} contributed by the molecule constituent atoms {X}, the key concept of the Orbital Communication Theory (OCT) of the chemical bond, is introduced and its information-theoretic (IT) bond descriptors are summarized. The additive and non-additive components of these molecular information channels are identified. The former involve only the internal (one-center) communications {X → X} between the basis functions χ X of each bonded atom X, determined by the associated (diagonal) block P(χ X|χ X) of the molecular conditional probabilities, which are responsible for the intra-atom promotion to its effective valence state. The latter accordingly involve only the external (two-center) communications between the contributed AO of each pair of bonded atoms, {Y → X and X → Y}, generated by the off-diagonal blocks of conditional probabilities P(χ X|χ Y) and P(χ Y|χ X),X ≠ Y, respectively, which are responsible for the inter-atomic bonding effects in the molecule. Both these probability scatterings ultimately determine the resultant multiplicities of the system chemical bonds. The input-ensemble average value of the channel conditional-entropy, which measures its communication “noise” due to electron delocalization via all chemical bonds, measures the IT-covalency in the molecule, while the complementary descriptor of the ensemble average value of the network mutual-information (information-capacity) reflects the electron localization effects and measures the system IT-ionic component. The illustrative example of the localized chemical bond originating from the interaction between two atomic orbitals is reexamined in some detail and the bond-weighted ensemble approach to chemical interactions in diatomic molecular fragments is discussed within the standard Restricted Hartree-Fock theory. In diatomic systems such treatment exactly reproduces the familiar bond index of Wiberg and provides its resolution into the complementary IT-covalent and IT-ionic components. The operator formulation of the probability-scattering phenomena in molecules is given and the probability-amplitude channel defined by the first-order density matrix is introduced. The AIM internal and external eigenvalue problems of this Charge-and-Bond-Order matrix are introduced and a similar approach to probability propagation matrices/operators is suggested.

Some formal comments on spin projected hartree-fock theory

Antisymmetric projection for no spin-orbit coupling is carried out within the one-electron approximation. A basis set is chosen such that one orbital of the minority spin space uniquely “pairs” with one and only one orbital of the majority spin space. The general expression for the expanded wave function given elsewhere [1] are shown to give rise to a relatively condensed, real expression for the system energy. The energy expressions can be separated into parts which one can argue in order: 1 Produce a strong tendency to minimum total spin and maximum overlap (double occupancy). 2 In the case of degenerate or nearly degenerate one-electron states there is a tendency to maximize the total spin value within this manifold (Hunds rule). However, it is expected that this tendency will be smaller than that calculated using restricted Hartree-Fock theory. 3 There is also a tendency opposing and producing some departure from complete overlap. All of these effects depend on terms whose individual elements can be expected to be equal to or larger than the usual exchange integrals.

A higher level ab initio quantum-mechanical study of the quadrupole moment tensor components of carbon dioxide

The quadrupolarity of carbon dioxide has been studied with higher level ab initio methods. Carbon dioxide exhibits {− + −} quadrupolarity in all directions and an explanation is provided of the origin of the sign of the diagonal elements Qii. The quadrupole moment tensor has been computed using restricted Hartree–Fock theory, second-order Møller–Plesset perturbation theory and quadratic configuration interaction theory. A variety of basis sets have been employed up to basis sets of the type [5s, 4p, 2d, 1f] (23s, 8p, 2d, 1f). The quadrupole moment tensor component Q∥ of carbon dioxide falls in the range between −18.5 and −20.5DebyeÅ. The quadrupole moment tensor components Q⊥ of carbon dioxide are smaller, ranging from −14.5 to −15DebyeÅ, and they are less sensitive to the choice of the theoretical model. The correlated methods consistently predict an increase of Q∥ while they predict a more modest reduction of Q⊥. It is for the opposing electron correlation effects on Q∥ and Q⊥ that the average values of the diagonal elements, 〈Qii〉, are essentially independent of the method and exhibit only a small variation depending on the basis set. On the other hand, the anisotropy of the quadrupolarity, the quadrupole moment Θ, is affected most by the opposing electron correlation effects on Q∥ and Q⊥. The accurate reproduction of the measured quadrupolarity Θ=−4.3DebyeÅ requires a theoretical model that employs both a good method and a good basis set. The results suggest that the use of second-order Møller–Plesset perturbation theory in conjunction with well-polarized triple-ζ basis sets provides a cost-effective and quite accurate method for the estimation of correlation effects on quadrupole moments.

Stereochemistry and Stereoelectronics of Azines. 13. Conformational Effects on the Quadrupolarity of Azines. An Ab Initio Quantum-Mechanical Study of a Lateral Synthon

The quadrupole moment of formaldazine, H2C=N-N=CH2, has been studied for the trans structure (Ð(C-N-N-C) = τ = 180≡) and a series of gauche structures (τ > 120≡). Restricted Hartree-Fock theory, second-order Moller-Plesset theory, and quadratic CI theory have been used in conjunction with the basis sets 6-31G*, 6-31G**, 6-311G** and 6-311++G**. Formaldazine is a quadrupolar molecule with primitive quadrupole moment tensor components of Qxx = -22.4, Qyy = -20.4 and Qzz = -25.6 DA at the theoretical level QCISD/6-311++G**. The examination of the theoretical level dependency shows that the reliable computation of a quadrupole moment requires the use of a flexible basis set. A large part of the component Qzz = -25.6 DA is due to the π-system and compares, on a per electron basis, with the Qzz value of benzene. Conformational changes of the azines in the range 120° < τ < 180≡ have but a minute effect on the energy and are associated with only minor electronic relaxation. These conformational changes alter the quadrupole moment tensor components less than ΔQxx = +0.4, ΔQyy = +1.6 and ΔQzz = -1.0 DA at QCISD/6-311++G**//QCISD/6-31G*. The direction of these changes is explained by consideration of the rotation of the CN-π-systems and a small reduction of the CN bond polarity in the gauche structures. The Qzz component of formaldazine is representative of the quadrupole moment tensor component along the direction of the C2 axis of the azine bridge as such. Hence, the results of this study suggest that azines can engage in strong quadrupole-quadrupole interactions and can be employed as lateral synthons in crystal engineering.

IR, Raman and theoretical ab initio RHF study of aminoglutethimide — an anticancer drug

A comparative analysis of the IR and Raman spectra of aminoglutethimide (AG) dissolved in CCl4, CHCl3 and CH3CN was performed. Most of the absorption bands were assigned to characteristic group vibrations with the use of the IR and Raman spectra of deuterated AG, glutethimide, N-methyl glutethimide and glutarimide. The AG samples very weakly interacting with the environment were studied with the use of the Ar matrix isolation IR spectra. For comparison, the IR and Raman spectra of the crystalline samples formed by hydrogen-bonded AG molecules were recorded. The spectra were analyzed also in terms of normal modes and the harmonic approximation with the use of the ab initio restricted Hartree-Fock theory. It was found that increasing the solute concentration in CCl4 and CHCl3 leads to formation of the autoassociates. In CH3CN the solute–solvent AG–CH3CN dimers occur. Possible structures of the associates were theoretically studied on the model systems: the centrosymmetric glutarimide dimer and the linear AG–CH3CN dimer. By a comparison of the theoretical and experimental spectra we were able to identify several peaks originating from the solute–solvent interactions.

Matrix elements of multi-electron operator in restricted Hartree-Fock theory

A theorem has been established for the matrix elements of a general t-electron operator between N-dimensional determinantal wave functions arising in the solution of the atomic and molecular multi-electron problem by the restricted Hartree-Fock (RHF) method (1 ≤ t ≤ N). The required matrix elements of this operator are sums of matrix elements over t-dimensional basic determinantal wave functions. The final results are especially useful in the determination of multi-electron properties for atoms and molecules when the Hylleraas approach in RHF theory is employed.

Elimination of methane from protonated acetaldehyde: The Ab initio transition state

The transition state (TS) for loss of CH 4 from protonated acetaldehyde has been located at the second-order Moller-Plesset (MP2)/6-31G( d,p) level of theory. The activation energy is predicted to be 263.9 kJ/mol starting from the more stable form (methyl and hydrogen E) and 261.6 kJ/mol starting from the less stable form (methyl and hydrogen Z) that is required for reaction. The products (methane and the formyl ion) are predicted to lie 136.6 kJ/mol below the TS for their formation. MP2 methods underestimate the heats of formation of both the TS and the reaction products by about 40 kJ/mol when compared with experiment. Restricted Hartree-Fock (RHF) calculations give much more accurate relative energies. The MP2 TS leads directly to fragmentation and is described as a protonation of the methyl group by the acidic proton on oxygen. Under RHF theory the reaction is stepwise. An RHF TS similar to the MP2 TS leads to a nonclassical intermediate (which is stable at this level of theory) that has one of the CH bonds protonated. This mechanism (protonation of an alkyl group) appears to be a general one for high energy 1,2 eliminations from organic cations. ( J Am Soc Mass Spectrom 1994, 5, 1102-1106)

Is a C C quadruple bond possible?

Ab initio computations predict a minimum for a carbon-carbon quadruple bond in C 2). While the computed C C separation, 1.128Å, fits Pauling's bond order-bond length relationship well, this result is an artifact of the restricted Hartree Fock theory.

On the ab initio determination of higher-order force constants at nonstationary reference geometries

Several complementary analyses have been performed in an investigation of the use of reference geometric structures which are not stationary at a given level of theory in the prediction of improved equilibrium anharmonic molecular force fields. Diatomic paradigms for the procedure were established by constructing empirical potential energy functions for the nitrogen and fluorine molecules which not only reproduce the available Rydberg–Klein–Rees data but also provide reliable derivatives through fourth order for ranges of 0.4 Å or greater around the equilibrium bond distance. For comparison, analogous curves were determined at the double-ζ plus polarization (DZP) restricted Hartree–Fock (RHF) level of theory, and the quartic force fields for N2 and F2 were also obtained at the experimental re structures using a (8s5p3d2f1g) basis set and the coupled-cluster singles and doubles method augmented by a perturbative contribution from connected triple excitations [CCSD(T)]. The results substantiate the ability of RHF theory to predict correlation-quality, higher-order force constants if an accurate reference geometry from experiment or a higher level of theory is employed. The theoretical foundations of this technique as applied to polyatomic molecular systems have been systematically explored. Mechanisms were analyzed which address the nonzero force dilemma by using various choices of internal coordinates to shift the equilibrium point of theoretical potential energy surfaces. Examples are presented in which the variations in predicted spectroscopic constants arising from different shift coordinate sets are non-negligible. A Cartesian projection scheme for higher-order force fields was developed and implemented to avert internal-coordinate dependences; formulas for higher-order projection matrices and higher-order derivatives of the external variables of a molecular system were concurrently derived. A formalism for the transformation of force fields between internal and Cartesian representations was also constructed which is applicable to arbitrary order. In addition to N2 and F2, case studies were performed on the F2O and N2O molecules, for which electron correlation effects are of unusual importance. Quartic force fields are reported for F2O and N2O at the DZP and TZ(2d1f) CCSD(T) levels of theory, respectively, which provide the best data sets currently available and facilitate the assessment of experimental force constants. The CCSD(T) results are reproduced remarkably well by RHF predictions at the experimental equilibrium structures of these molecules but not at the corresponding RHF optimum geometries. Finally, practical recommendations are made for predictions of higher-order force constants at nonstationary points.

Kinetic and Exchange Energy Densities of an Inhomogenous Electron Liquid Related in Hartree—Fock Theory and Other Models

Abstract March and Santamaria have recently proposed non-local generalizations of kinetic and exchange energy densities of an inhomogeneous electron liquid and have related these through restricted Hartree–Fock theory. Prompted further by the work of Lee, Lee and Parr, the above results are generalized to include the unrestricted Hartree–Fock theory. Finally, approximations of the type required to relate to the work of Lee et. al. are referred to.

Geometries, force fields, and vibrational assignments of Dewar benzene and Dewar pyridine

The geometries of Dewar benzene and Dewar pyridine were fully optimized by restricted Hartree-Fock theory and the 6-31 G* basis set. Reference geometries for force field calculations were obtained by systematic empirical corrections. The ab initio force fields, after empirical scaling, yield frequencies which are in good agreement with experiments. On the basis of the calculations, some uncertainties in the experimental assignments of the fundamental frequencies of Dewar benzene were resolved, and assignments for the observed matrix IR frequencies of Dewar pyridine were given for the first time

The infrared spectrum of trimethylenemethane. Predictions of in-plane vibrational frequencies from correlated wave functions

The infrared vibrational spectrum of trimethylenemethane (TMM) is predicted with self-consistent field and configuration interaction methods. The 3A′2 electronic ground state of TMM is described in terms of restricted Hartree–Fock theory and in light of experimental evidence. Analytic gradient methods are employed to optimize theoretical geometries for 3A2 TMM; vibrational frequencies are evaluated via analytic second-derivative techniques (self-consistent-field) and finite differences of analytic gradients (configuration interaction). The resulting IR-spectral predictions are modified to reflect average errors statistically associated with the two theoretical methods.

A generalized restricted open-shell Fock operator

We reexamine the open shell restricted Hartree-Fock theory and develop Fock-like operators that are quite general and easy to implement on a computer. We present a table of ‘vector coupling coefficients’ that define this operator for most of the cases that commonly arise. We compare the form of this operator with that suggested by others, and discuss the orbitals obtained by this procedure with respect to the generalised Brillouin's theorem, and the orbital energies with respect to Koopmans' approximation.

Study of the orientation of thiourea adsorbed on aluminum oxide by tunneling spectroscopy. I. Determination of partial charges from molecular electrostatic potential calculations for thiourea and urea.

Ab initio electrostatic molecular potentials for urea [CO(${\mathrm{NH}}_{2}$${)}_{2}$] and thiourea [CS(${\mathrm{NH}}_{2}$${)}_{2}$] have been calculated with use of restricted Hartree-Fock theory. A nonlinear least-squares regression package was used to fit atomic partial charges to the potentials. Calculations were made with use of several different basis sets with and without polarization functions. Basis-set scaling factors calculated for each atom showed that the potential-derived (PD) charges were much more reliable than the Mullikan values. The effects of a vibrational mode on the calculated partial charges were also evaluated. Dipole moments were calculated with use of various basis sets and compared with observed values. The potential was also fitted by splitting the static charge determined for each atomic site into two portions. The core-charge portion was forced to follow the nuclear motion, whereas the valence-charge portion was allowed to float in a direction such that the best fit to the potential was obtained. This latter procedure provided reproductions of the calculated quantum-mechanical electrostatic potentials which were determined to be sufficiently accurate for electron-tunneling-intensity calculations. Comparison of experimental and calculated tunneling intensities for thiourea are presented in the following paper.