The relationship between the unrestricted and projected Hartree–Fock methods in a simple three-electron model system

Various forms of the unrestricted Hartree–Fock (UHF ) scheme are studied for a simple three-electron model system, represented by the PPP (Pariser–Parr–Pople) π-electron model of the allyl radical. Both spin and space symmetry are violated in the UHF trial wave function, either individually or simultaneously. A comparison with the projected Hartree–Fock (PHF ) schemes studied earlier is made and the effect of the order in which various symmetries are broken in both UHF and PHF schemes is studied. The effectiveness of various schemes follows from a comparison of the correlation energy and the wave function is obtained by various UHF (or projected UHF ) and PHF schemes, in the whole range of the coupling constant, with the corresponding quantities given by the exact solution of the model. Finally, the implications of the stability of the restricted HF solutions for the behavior of various single- and multiparameter UHF and PHF schemes are briefly outlined and exemplified on the studied model.     

On the convergence of the Møller-Plesset perturbation series

Møller-Plesset perturbation energies, up to as much as 48th order, have been calculated for H₂O (RHF and UHF framework) and NH₂ (UHF framework) atC _2v geometries (r _e , 1.5r _e and 2r _e ). Atr _e , the RHF and UHF series rapidly converge, but at2r _e , the RHF series converges erratically with an energy at 43rd order within 10^–7 hartree of the exact value whereas the UHF series converges smoothly, but very slowly, and for H₂O has an error in excess of 10^–5 hartree at 48th order. The significance of these results is discussed.     

Chiral Discrimination in Dimers of Diphosphines PH2PH2 and PH2PHF

A theoretical study of the conformational profile of two diphosphines, PH₂?PH₂ and PH₂?PHF, is carried using second‐order Møller–Plesset perturbation theory (MP2) computational methods. The chiral minima found are used to build homo‐ and heterochiral dimers. Six minima are found for the (PH₂?PH₂)₂ dimers and 27 for the (PH₂?PHF)₂ dimers. Pnicogen and hydrogen bonds, the non‐covalent forces that stabilize the complexes, are characterized by Atoms in Molecules (AIM) and Natural Bond Orbital (NBO) methodologies. Those with several pnicogen bonds are more stable than those with hydrogen bonds. The chirodiastaltic energies amount to a total of 1.77 kJ mol^?1 for the R_a:R_a versus R_a:S_a (PH₂?PH₂)₂ dimers, 0.81 kJ mol^?1 for the RS_a:RS_a versus RS_a:SR_a (PH₂?PHF)₂ dimers, and 2.93 kJ mol^?1 for the RR_a:RR_a versus RR_a:SS_a (PH₂?PHF)₂ dimers. In the first and second cases, the heterochiral complex is favored whereas in the third case, the homochiral complex is favored.     

Ab initio study of reactions of hydroxyl radicals with chloro‐ and fluoro‐substituted methanes

Ab initio calculations at the unrestricted Hartree–Fock (UHF) level have been performed to investigate the hydrogen abstraction reactions of ^? OH radicals with methane and nine halogen‐substituted methanes (F, Cl). Geometry optimization and vibrational frequency calculations have been performed on all reactants, adducts, products, and transition states at the UHF/6‐31G* level. Single‐point energy calculations at the MP2/6‐31++G* level using the UHF/6‐31G* optimized geometries have also been carried out on all species. Pre‐ and postreaction adducts have been detected on the UHF/6‐31G* potential energy surfaces of the studied reactions. Energy barriers, ΔE^?, reaction energies, ΔE_r, reaction enthalpies, ΔH_r, and activation energies, E_a, have been determined for all reactions and corrected for zero‐point energy effects. Both E_a and ΔH_r come into reasonable agreement with the experiment when correlation energy is taken into account and when more polarized and diffuse basis sets are used. The E_a values, estimated at the PMP2/6‐31++G* level, are found to be in good agreement with the experimental ones and correctly reproduce the experimentally observed trends in fluorine and chlorine substitution effects. A linear correlation between E_a and ΔH_r is obtained, suggesting the presence of an Evans–Polanyi type of relationship. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 426–440, 2001     

Half-Projected and Projected Hartree-Fock Calculations for Singlet Ground States. i. four-Electron Atomic Systems

The half-projected Hartree-Fock function (HPHF ) for singlet states is defined as a linear combination of two Slater determinants, which contains only spin eigenfunctions with even quantum number. Using a self-consistent procedure based on the generalized Brillouin's theorem, the RHF , HPHF and PHF functions are deduced for the ground states of the Li^?, Be, B⁺, and C²⁺ systems, in a limited basis set. It is found that the HPHF function yields better energy values than the RHF function, very close to that of the PHF one. The HPHF scheme seems thus to be useful as a substitute for the PHF model, specially in the case of large electronic systems in which the latter method becomes unmanageable.     

Ab initio study of possible intermediates in the no(4Π) catalyzed geometric isomerizations of olefins

The geometries of acyclic and three-membered ring (nitroxide) H₄C₂NO radicals in their ground ²Π electronic states have been optimized completely at ab initio UHF and ROHF theoretical levels with the STO-3G and the 6-31G** basis sets. The optimizations favour the cyclic nitroxide structure energetically. However ΔE(acyclic - cyclic) at the UHF and ROHF/6-31G** levels are only 3.2 and 1.9 kcal mol^-1, respectively. Incomplete MP2/6-311G** optimizations support these results. The zero-point energy computed at the ROHF/6-31G** level for the nitroxide radical is 2.5 kcal mol^-1 higher than that for the acyclic structure, thus reversing the relative energies by 0.6 kcal mol^-1. The energies of the two radical structures, relative to the sum of those for ethylene and NO, are very close to literature values of the activation energies for the thermal, NO catalyzed geometrical isomerizations of olefins. Thus cyclic nitroxide intermediates may play a role not only in the Hg 6(³P₁) photosensitized, but also in the thermal, NO-catalyzed geometric isomerizations of olefins. Paper dedicated to Professor Otto P. Strausz; presented in part at the 75th Canadian Chemical Congress and Exhibition, Edmonton, May 31 – June 4, 1992.     

Intramolecular Pnicogen Interactions in PHF?(CH2)n?PHF (n=2–6) Systems

A computational study of the intramolecular pnicogen bond in PHF? (CH₂)_n? PHF (n=2–6) systems was carried out. For each compound, two different conformations, (R,R) and (R,S), were considered on the basis of the chirality of the phosphine groups. The characteristics of the closed conformers, in which the pnicogen interaction occurs, were compared with those of the extended conformer. In several cases, the closed conformations are more stable than the extended conformations. The calculated interaction energies of the pnicogen contact, by means of isodesmic reactions, provide values between ?3.4 and ?26.0 kJ mol^?1. Atoms in molecules and electron localization function analysis of the electron density showed that the systems in the closed conformations with short P ??? P distances have a partial covalent character in this interaction. The calculated absolute chemical shieldings of the P atoms showed an exponential relationship with the P ??? P distance. In addition, a search in the Cambridge crystallographic database was carried out to detect those compounds with a potential intramolecular pnicogen bond in the solid phase.     

He2+2: A comparison between Hartree–Fock and density functional techniques

A comparison of Roothaan-Hartree–Fock methods (both restricted and unrestricted) with density functional ones (LCAO -Xα and cellular MS -Xαβ) is made using as test case the He²₂ + molecular ion. It is shown the analogy that exists between RHF and symmetry-adapted LCAO -Xα potential energy curves, as well as between UHF and symmetry-unconstrained LCAO -Xα ones. The influence of symmetry adaptation on the overall behavior of the potential energy curve is also discussed. Finally, the difference in the behavior of the LCAO -Xα and cellular MS -Xαβ calculations is explained as an artifact of the space partitioning in the latter technique. It is concluded that LCAO -Xα method is superior to cellular MS -Xαβ because it requires less effort to reach the same results and that the general behavior is similar to UHF , although the former affords a better equilibrium bond distance and a worse energy barrier than the latter.     

The restriction of spin contamination in unrestricted Hartree Fock wave functions

A method is developed to study in detail the contaminating spin components in UHF calculations for free radicals. Applications are made to those cases previously reported in which normal parametrisation of PPP UHF calculations lead to values of S ² which increase on annihilation of the contaminating quartet component.

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Zusammenfassung Es wird eine Methode entwickelt, um detaillierte Untersuchungen der bindenden Spinkomponenten in UHF-Rechnungen für freie Radikale zu ermöglichen. Es werden Anwendungen für solche Fälle, über die schon früher einmal berichtet wurde, durchgeführt, bei denen die normale Parametrisierung von PPP-UHF-Rechnungen zu Werten von S ² führte, welche die Vernichtung der bindenden Quartettkomponente anwachsen lassen.

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Résumé Développement d'une méthode pour étudier en détail les composantes de spin étrangères dans les calculs UHF des radicaux libres. On l'applique aux cas précédemment considérés dans lesquels des calculs UHF avec paramétrisation PPP conduisent à des valeurs de S² qui croissent lorsque l'on annule la composante quartet.

     

Rapid estimation of activation enthalpies for cytochrome-P450-mediated hydroxylations

Cytochrome P450 (CYP) enzymes play a critical role in detoxication and bioactivation of xenobiotics; thus, the ability to predict the biotransformation rates and regioselectivity of CYP enzymes toward substrates is an important goal in toxicology and pharmacology. Here, we present the use of the semiempirical quantum chemistry method SAM1 to rapidly estimate relative activation enthalpies (ΔH^?) for the hydroxylation of aliphatic carbon centers of various substrates. The ΔH^? were determined via a reaction path calculation, in the reverse direction (RRP), using the iron‐hydroxo‐porphine intermediate and the substrate radical. The SAM1 ΔH^? were calculated via unrestricted Hartree‐Fock (UHF) and configuration interaction (CI) formalisms for both the doublet and quartet spin states. The SAM1 RRP ΔH^?, after subtracting a correction factor, were compared with density functional theory (DFT) B3LYP activation energies for two sets of substrates and showed R² ranging from 0.69 to 0.89, and mean absolute differences ranging from 1.2 ± 1.0 to 1.7 ± 1.5 kcal/mol. SAM1 UHF and CI RRP calculation times were, on average, more than 200 times faster than those for the corresponding forward reaction path DFT calculations. Certain key transition‐state (TS) geometry measurements, such as the forming O···H bond length, showed good correlation with the DFT values. These results suggest that the SAM1 RRP approach can be used to rapidly estimate the DFT activation energy and some key TS geometry measurements and can potentially be applied to estimate substrate hydroxylation rates and regioselectivity by CYP enzymes. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Ab initio studies of tetrafluoroethylene and tetratrifluoromethylethylene molecules in the ground and “twisted” states

Ab initio calculations with full optimization of geometry have been carried out with the 6–31 G^* basis set on tetrafluoroethylene (with the unrestricted Hartree-Fock method—UHF and the second-order Moller-Plesset perturbation theory—MP2) and tetratrifluoromethylethylene (with UHF) molecules in the singlet ground and triplet biradical states. The symmetry of the tetrafluoroethylene molecule in the triplet biradical state was demonstrated to differ from that of ethylene (D _2d ) due to the deviation of fluorine atoms from CCFF plane. The MP2 optimized geometries of ethylene and tetrafluoroethylene were used for higher level calculations (MP3, MP4, CCSD). The energy of the ground state singlet-biradical triplet splitting decreases in the series: ethylene>tetrafluoroethylene> tetratrifluoromethylethylene. These data on energy splitting explain the increase in reactivity toward the [2+2]-cycloaddition on going from ethylene to tetrafluoroethylene. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 605–607, April, 1998.     

Theoretical studies on quinones I. The structure of p-benzoquinone and two of its excited triplet states

Ab initio calculations on the ground and two excited triplet states (³B_1g and ³B_1u) of p-benzoquinone are described. The geometries of the three states were fully optimised at the SCF level using the 3-21G basis set. For the excited states, both D_2h and C_2v geometries were investigated. Comparison was made between UHF and ROHF levels of theory.     

An average fock operator technique of approximate open-shell LCAO-MO-SCF calculation

The exchange part of the usual Hartree-Fock potential in the unrestricted Hartree-Fock (UHF) theory is suitablyaveraged to construct an, average one-electron model Hamiltonian which generates a set of spin-restricted one-electron orbitals in a self-consistent manner. These orbitals are then used to calculate the electronic energy of the open-shell system by using the proper functional form for the energy which handles the exchange terms correctly. The eigenvalues ofF ^av can be used for calculating either the spin-polarised or spin-averaged ionisation potentials of different orbitals at theKoopmans’ theorem level of approximation. Comparison ofE ^ac with the UHF-energy shows thatE _UHF<E ^ac in each case revealing some kind of an upper bound nature ofE ^ac. An approximate variational argument is given. Relationship of our model with the hyper-Hartree-Fock theory of slater is explored and the general problem of eliminating ‘self-interaction’ terms in average Fock-operator based theories is discussed.     

Indo study of the angular dependences of isotropic hyperfese interaction constants in the model conformations of the nitrobenzene radical anion

The UHF/INDO calculations of the model conformations of the nitrobenzene radical onion show that rotation of the nitro group relative to the plane of the benzene ring is accompanied by a pyramidal distortion of the group caused by the pseudo-Jahn-Teller effect (vibronic interaction between the ground n and totally symmetric lowest excited σ states). The angular dependences of the¹⁴N,¹³C,¹H, and¹⁷O Isotropic hyperfine interaction constants are analyzed. Experimental ESR data are interpreted for the radical anions of nitrobenzene derivatives with ortho-alkyl groups. Translated fromZhumal Strukturnoi Khimii, Vol. 41, No. 3, pp. 457-467, May-June, 2000.     

Multiple solutions of unrestricted Hartree-Fock equations: The SNH+ radical as an example

A case of appearance of multiple unrestricted Hartree-Fock (UHF) solutions in the SNH⁺ (²A') radical is reported. This kind of solution does not arise from different electron configurations or symmetry breaking effects and seem to be exclusive of the UHF method. Three solutions are found and studied on the basis of wave function stability tests and several conclusions about their origin are given.     

Electronic structure of long cumulene chains

The analysis of the equations of the unrestricted Hartree–Fock (UHF) method for polyenes C_NH_N+2 with even and odd N » 1 is carried out. The equations of the UHF method are shown to be the same in both cases. The comparison of the UHF method with the extended Hartree–Fock (EHF) method applied to large systems is performed. The ground state and π-electron spectra of long cumulene chains C_NH₄ are treated by the EHF Method. The end effects are taken into consideration. It is shown that the EHF method gives a finite value of the first optical transition frequency and, at the same time, zero value of torsion barrier of end CH₂–groups in long cumulene chains (N → ) in contrast to previous calculations of cumulenes by the Huckel method and the restricted Hartree–Fock method.     

The Hartree-Fock dissociation of F2

We examine the stability of the restricted Hartree-Fock (RHF) wave function for F₂ in the vicinity of the equilibrium internuclear distance (R=R _e ) and the shape of the unrestricted Hartree-Fock (UHF) potential energy curve for the same system. The results depend on the basis set: With a split valence plus polarization basis, 6-31G(d), the RHF wave function is unstable at R _e , and the UHF potential curve is purely dissociative. When the basis is extended to 6-311+G(3d) or 6-311+G(3df), the RHF wave function becomes stable, and the UHF potential curve acquires a local maximum for R slightly (0.02 å) greater than R _e . The local maximum, however, is only 0.1 kcal/mol higher than the local minimum at R=R _e .     

Structure and bonding study of the P2 + P2+ system

The structures of P₂ + P₂⁺ were studied with ab initio calculations at the ROHF and UHF levels with the 6-31G* basis set. The geometries and dissociation energies for the four selected structures–collinear, T-shaped, regular trapezoid, and elongated tetrahedral–were studied in comparison with N₂ + N₂⁺. The trade-off of the intramolecular π bond for the intermolecular σ bond for the P₄⁺ system results in its larger dissociation energies and more substantial changes in bond distances than those in the N₄⁺ system. © 1995 John Wiley & Sons, Inc.     

室温离子液体FeCl3-BPC体系的研究

利用差示扫描量热法(DSC)建立了无水三氯化铁和氯化正丁基吡啶(BPC)二元体系相图. 依据相图, FeCl₃和BPC形成室温离子液体的窗口是x＝0.26～0.58; 室温离子液体的深度是80 ℃. 利用UHF/6-31G^*对FeCl₃, FeCl₄^－, Fe₂Cl₇^－等配合物的几何结构、键长、能量和Raman频率进行优化, 从头算和Raman光谱证实了相图中FeCl₃摩尔分数x＝0.50处有稳定化合物存在, FeCl₄^－是主要阴离子; x＝0.67处, FeCl₄^－, Fe₂Cl₇^－是主要阴离子.