Restricted Hartree-Fock Solutions Research Articles

CCSD(T) calculations of confined systems: in-crystal polarizabilities of F⁻, Cl⁻, O²⁻, and S²⁻.

We explore dipole polarizabilities of the singly and doubly charged anions F(-), Cl(-), O(2-), and S(2-) in an external, harmonic oscillator (HO) confining potential ∑(i)½ω(2)r(i)(2). We find that in contrast to F(-) and Cl(-) those for O(2-) and S(2-) are unrealistically high due to the instability of the corresponding restricted Hartree-Fock (RHF) solutions. Yet, already a relatively weak HO confining potential stabilizes their RHF solutions and eliminates any possible broken-symmetry solutions. The coupled-cluster theory with single, double and noniterative triple excitations (CCSD(T)) then yields considerably reduced polarizabilities for O(2-) and S(2-) relative to their unconfined values. We showed that polarizabilities of O(2-) and S(2-) are more sensitive to the strength of a confinement potential than are those for F(-) and Cl(-). This enables us to relate the confining parameter ω with the known experimental polarizabilities for selected crystals (our "training set") and to find a specific confining parameter ω for which the CCSD(T) polarizability equals the experimental in-crystal polarizability of an anion in the training set. The latter may then be used as an alternative approach for determining the in-crystal polarizabilities of anions by exploiting the fact that the characteristic ω values depend linearly on the ionic radius of a cation participating in specific crystals containing these anions. Using this method we then calculate the isotropic dipole polarizabilities for F(-), Cl(-), O(2-), and S(2-) embedded in the LiF, LiCl, NaF, NaCl, KF, KCl, ZnO, ZnS, MgO, MgS, CaO, CaS, SrO, SrS, BaO, BaS, and other crystals containing halogen, oxygen, or sulphur anions. We compare our results with those obtained via alternative models of the in-crystal anionic polarizabilities.

Do independent-particle-model broken-symmetry solutions contain more physics than the symmetry-adapted ones? The case of homonuclear diatomics

We explore spin-preserving, singlet stability of restricted Hartree-Fock (RHF) solutions for a number of closed-shell, homonuclear diatomics in the entire relevant range of internuclear separations. In the presence of such instabilities we explore the implied broken-symmetry (bs) solutions and check their stability. We also address the occurrence of vanishing roots rendered by the stability problem in the case of bs solutions. The RHF bs solutions arise primarily due to the symmetry breaking of the relevant, mostly frontier, molecular orbitals, which approach atomic-type orbitals in the dissociation limit. The resulting bs RHF solutions yield more realistic potential energy curves (PECs) than do the symmetry adapted (sa) solutions. These PECs are shown to be very similar to those rendered by the density functional theory (DFT). Moreover, the sa DFT solutions are found to be stable in a much wider range of internuclear separations than are the RHF solutions, and their bs analogs differ very little from the sa ones. Finally, we examine a possible usefulness of bs RHF solutions in post-HF correlated approaches to the many-electron problem, specifically in the limited configuration interaction and coupled-cluster methods.

Symmetry breaking in spin-restricted Hartree–Fock solutions: the case of the C2 molecule and the N2+ and F2+ cations

We examine a spin-preserving stability of restricted Hartree-Fock (RHF) and open-shell RHF (ROHF) solutions for homonuclear diatomic species, namely the molecule C(2) and the N(2)(+) and F(2)(+)cations, in the entire relevant range of internuclear separations. In the presence of respective singlet, doublet, or triplet instabilities we explore the implied broken-symmetry (BS) solutions and check their stability. We also address the occurrence of vanishing roots rendered by the stability problem in the case of BS solutions. Since for homonuclear diatomic species the space symmetry of the nuclear framework cannot be broken, the only possible space symmetry breaking involves either the electron charge distribution or the rotational symmetry about the internuclear axis. Such BS solutions arise primarily due to the symmetry breaking of the valence, mostly frontier, molecular orbitals, which approach atomic-type orbitals in the dissociation limit. The resulting BS RHF or ROHF solutions yield much more realistic potential energy curves (PECs) than do the symmetry-adapted (SA) solutions. For the sake of comparison, we also generated corresponding PECs using the density functional theory (DFT). Finally, we examine the role of BS molecular orbitals in post-HF correlated approaches to the many-electron problem, specifically in the computation of equilibrium properties using the coupled-cluster method with singles and doubles (CCSD) and its perturbatively corrected version for triples, the CCSD(T) method.

Locating Multiple Self-Consistent Field Solutions: An Approach Inspired by Metadynamics

We propose a method to locate the solutions to the self-consistent field (SCF) equations, using an approach based upon metadynamics. We define a distance function between density matrices. Within an SCF calculation, when a solution is found, a biasing potential based on distance from the solution is added to the energy to avoid reconvergence to the same solution. Multiple solutions can therefore be relatively easily found. Using this method we locate all known solutions and one unknown solution of the H4 model. The set of restricted Hartree-Fock (RHF) solutions for the nitrogen molecule is located, and a broken-symmetry solution lower in energy than the symmetric RHF solution is found corresponding to dissociation into doublet fragments.

Independent particle model of spontaneous symmetry breaking in planar π-electron systems

The singlet stability of symmetry adapted (SA), restricted Hartree-Fock (RHF) solutions, and the implied symmetry breaking for several planar, π-electron systems, is investigated using the semiempirical Pariser-Parr-Pople Hamiltonian in the whole range of the coupling constant. We focus here on highly symmetric cyclic polyenes C10H10 and C14H14 and their various distorted analogues of lower symmetry, in particular on the perimeter models of naphthalene and anthracene (p-naphthalene and p-anthracene) modeling the so-called [n]-annulenes. Relying on earlier results for general systems with conjugated double-bonds, we explore the character and properties of both the SA and broken-symmetry (BS) RHF solutions for these systems and relate their behavior to those of highly symmetric cyclic polyenes and corresponding polyacenes. In this way we are able to provide a better understanding of the spontaneous symmetry breaking in these systems at the Hartree-Fock level of approximation.

Negative Conductance in Coulomb Blockade

Based on the Anderson model with nonequilibrium distribution, we discuss the current between two electrodes through an atom with a finite bias voltage. The Anderson model is used to study the Coulomb blockade phenomenon from the small system size limit. We employ the Mahanthappa-Baksi-Keldysh Green function formalism to deal with the nonequilibrium system. The second-order perturbation study with respect to the Coulomb interaction between electrons in the atomic orbital is performed by starting with the restricted Hartree-Fock solution. We show results of various cases of the relative position of energy level on the atom with respect to the Fermi energies in two electrodes. Several behaviors of the conductance, such as the Kondo-assisted tunneling and blocking of the current by the Coulomb repulsion, are found. It is also shown that the conductance can be negative under special conditions. We discuss the physical nature of these effects in terms of the local density of states on the atom.

Aromaticity of annulenes and annulene ions with 4v+2 electrons from the viewpoint of the theory of Hartree-Fock instabilities

The connection between the aromaticity of annulenes and annulene ions with 4v+2 π-electrons and the stability of the closed-shell restricted Hartree-Fock (RHF) solutions for these systems is discussed in the framework of the PPP-approximation. It is shown that the tendency towards an uniform electron density distribution in aromatic cycles is paralleled by the stability properties of the corresponding closed-shell RHF solutions. The stability investigations are demonstrated to provide realistic estimates of the critical ring sizes at which Huckel's 4v+2 rule breaks down.

The Electronic Structure of the Fulvalene Trication Radical. A Localization of the Unpaired Electron

Abstract The restricted Hartree-Fock (RHF) solution for the trication radical of fulvalene, obtained by using the open-shell SCF formalism of the Pariser-Parr-Pople-type MO method, is found to be doublet unstable at the conventional D2h nuclear arrangement. We calculate the broken-symmetry charge-density wave solution lower in energy than the usual symmetry-adapted RHF solution and examine its properties. It is predicted that the unpaired electron should be localized on the single five-membered ring having a smaller positive charge. Finally, on the basis of the second-order Jahn-Teller effect, we examine the molecular-symmetry reduction of the trication radical from the fully-symmetrical D2h to the C2v nuclear arrangement.

Stability properties of closed‐shell restricted Hartree–Fock solutions for electronic systems in the framework of the projected Hartree–Fock method and their utilization

AbstractThis paper contains a derivation of stability conditions for closed‐shell restricted Hartree–Fock (RHF) solutions in the framework of the projected Hartree–Fock (PHF) method. It is shown that for any projection operator whose choice is consistent with the symmetry of the RHF solution, one obtains four stability subproblems (one “singlet” and three “nonsinglet”). The stability conditions are specialized to several particular choices of the projection operator, including Löwdin's spin projection operator for the singlet case (this choice corresponds to the celebrated spin‐PHF method). The stability of closed‐shell RHF solutions for alternat π‐electronic systems in the framework of the alternant molecular orbital (AMO) method is also discussed. The theoretical considerations are illustrated by PPP calculations for several closed‐shell π‐electronic models. The results for even alternant conjugated hydrocarbons show that the AMO construction is suggested by the structure of the eigenvector pointing into the direction of steepest descent from the pertinent RHF solution in the framework of the spin‐PHF method. The minimization of the spin‐PHF energy expectation value in this direction in a number of cases allows that value to approach rather closely the energy values obtained in a fully variational spin‐PHF treatment and thus provides either a one‐parameter variational method which is shown to be superior to the one‐parameter AMO method or a convenient initial approximation to the solution of the spin‐PHF equations.

Instability of the Hartree-Fock solutions for various conjugated heterocyclic systems

The regular electronic-structural features which were previously observed for a number of conjugated hydrocarbons, and which are due to the instability of the solutions in the restricted Hartree--Fock (RHF) approximation, have made it possible to distinguish various heterocyclic systems characterized by singlet and triplet instability of the RHF solutions. The problems of the stability and multiplicity of the RHF solutions for conjugated heterocyclic systems such as 1,4-dihydroquinoxaline-5,8-dione and its hetero-analogs, and also for aza derivatives of cyclobutadiene, have been examined. In the calculation by the PPP method, two different RHF solutions have been obtained, depending on the initial density matrix, corresponding either to a structure with delocalized ~ bonds or to a structure with localization of the ~ bonds. The singlet and triplet instability of a number of the solutions obtained has been discussed. The regular electronic-structural features due to the multiplicity of the solutions have been examined. The possibility of a triplet ground state for 1,4-dihydroquinoxaline-5,8-dione and some of its hereto-analogs has been suggested.

On the behaviour of exchange in restricted hartree-fock-roothaan calculations for periodic polymers

Analysis of the restricted Hartree-Fock (RHF) exchange potential is presented. In polymeric chains, a relation between the short-rangeness of the exchange and the energy gap is observed. In metallic systems, the exchange is of long-range and correlation corrections are expected to be important. The validity of RHF solutions in metallic situations is considered.

Hartree-fock instability in connection with the structure and geometry of some conjugated molecules

The electronic-structural regularities producing instability and multiplicity of restricted Hartree—Fock (RHF) solutions have been considered. Singlet and triplet instabilities of RHF solutions for some heterocyclic conjugated molecules have been found. The significance of the starting point in the SCF iterative procedure is examined. The existence of a triplet, or probably a biradical, ground state of 1,4-dihydroquinoxalinedione-5,8 and some of its heteroanalogues is suggested.

The doublet instabilities of the Hartree-Fock solutions for the cation and anion radicals of fulvalene systems

The doublet instabilities of the restricted Hartree-Fock (RHF) solutions for the ion radicals of fulvalene systems, some of which have been known to show the lattice instabilities, are examined in the framework of the semiempirical SCF MO approximation. The RHF solutions for the anion radical of heptafulvalene and the cation radical of pentafulvalene are found to be doublet unstable at the conventional D2h nuclear arrangement. We calculate the broken-symmetry charge-density-wave (CDW) solutions lower in energy than the usual symmetry-adapted RHF solutions and examine their properties. It is shown that the energy lowerings due to the CDW solutions are very small in both the ion radicals, but their density matrix elements are remarkably different from those for the symmetry-adapted RHF solutions. It is also shown that the proton hyperfine splittings for the anion radical of heptafulvalene calculated assuming the CDW solution are in good agreement with the experimental values. Finally, we refer to the relationship between the doublet instability of the RHF solution and the lattice instability.

Instability of the restricted Hartree-Fock (RHF) solution for the triplet state

Instability conditions of the restricted Hartree-Fock (RHF) solutions for the triplet state are investigated in connection with the orbital degeneracy of the H4 system of the D 4h , Td and D 3h symmetries. The relative stabilities of the RHF solutions and of the solutions obtained from the instability conditions are discussed. The correlation effects involved in the ground Hartree-Fock solutions are also discussed systematically. It is shown that the instability conditions of the RHF solutions are useful in investigating the correlation effects for the triplet species as well as singlet and doublet species.

The instability conditions of the restricted Hartree—Fock (RHF) solutions in the doublet state

Instability conditions of the restricted Hartree—Fock (RHF) solutions in the doublet state are investigated in connection with the orbital degeneracy for the case of the dissociation reaction of H 2 +, H 2 − and linear symmetric H 3 radicals. The instability of the RHF solution is also confirmed by a comparison between the total energies for the restricted and unrestricted HF (UHF) solutions.