Hartree-Fock Ground-state Energy Research Articles

Utilizing the Hartree-Fock Method to Analyze Carbon Dioxide Molecule

The determination of the ground state of the carbon dioxide molecule is conducted through matrix Hartree-Fock computations for a polyatomic molecule. Utilization of basis sets is implemented to achieve a precision that approaches the sub-μ Hartree of the CO<sub>2</sub> molecule for the energy levels. Utilizing a 28s14p14d14f atom-centered and a 24sl0plld bond-centered Gaussian basis set, the upper limit for the Hartree-Fock ground state energy of CO<sub>2</sub> at its practical picture has been calculated, yielding a value of -187.125408 Hartree. Utilizing our knowledge in diatomic molecules, we approximate the precision of this measurement to fall within 5-6 μ Hartree. The current computations offer a methodology to assess the precision of different basis sets frequently utilized in molecular self-consistent field investigations. These basis sets encompass STOJG, 4-31G, 6-31G, 6-31G(3d), 6-31IG, 6-311+G(3d0, D95, and D95V+(3d)), alongside cc-pVDZ, uug-cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ, which have been proposed for the purpose of investigating electron correlation. We have recently performed calculations on the CO1 ground state at the nuclear geomevy employed in the presenl work using the basis sets proposed by Dunning and co-workers and designated cc-pVQZ. aupcc-pVQZ, cc-pVSZ and nupcc-pVSZ. The estimated basis set truncation emrs for these sets are 0.00371, 0.003 19, 0.00047 and 0.00039 Hartree respectively.

Local magnetic moments due to loop currents in metals

We present Hartree-Fock calculations on a simple model to obtain the conditions of formation of local magnetic moments due to loop currents ${L}_{o}$ and spin-loop currents ${L}_{s}$ and compare them to the conditions of formation of local spin moments $M$ which were given long ago in a similar approximation by Anderson. A model with three degenerate orbitals sitting on an equilateral triangle, with on-site and nearest-neighbor repulsions $U$ and $V$, respectively, and intersite kinetic energy, hybridizing with conduction electrons with a parameter $\mathrm{\ensuremath{\Delta}}$ is investigated. ${L}_{o}$ and ${L}_{s}$ are promoted by large $V/\mathrm{\ensuremath{\Delta}}$ and their magnitude is relatively unaffected by $U/\mathrm{\ensuremath{\Delta}}$. Spin-magnetic moments $M$ promoted by large $U/\mathrm{\ensuremath{\Delta}}$ on the other hand are adversely affected by $V/\mathrm{\ensuremath{\Delta}}$. In this model, ${L}_{o}$ for $V$ multiplied by the number of neighbors is approximately the same as the $M$ promoted by $U$ in Anderson's local model for $M$. ${L}_{o}$ and ${L}_{s}$ are degenerate if exchange interactions and Hund's rule are neglected but ${L}_{o}$ is favored when they are included. Many of the qualitative results are visible in an expression for the Hartree-Fock ground state energy derived as a function of small ${L}_{o}, {L}_{s}$, and $M$. Numerical minimization of the Hartree-Fock energy is presented for larger values. We also briefly discuss the connection and differences of the interaction-generated orbital currents and spin currents discussed here and generalized to a lattice with the topological states in metals and semiconductors.

Hartree-Fock-like partitioning of the two-matrix for an exactly solvable two-electron model atom

Using a Hartree-Fock-like partitioning of the two-matrix in terms of the true correlated one-matrix, the corresponding approximate ground-state energy E{sub 0} is calculated for an exactly solvable closed-shell system proposed earlier by Moshinsky [Am. J. Phys. 36, 52 (1968)]. Comparisons of E{sub 0} with the Hartree-Fock ground-state energy E{sub HF} and the exact ground-state energy E{sub ex} are made. These comparisons provide valuable insight into the usefulness of a Hartree-Fock-like partitioning of the two-matrix with repulsive interparticle potentials. In particular, an explicit quantification for the applicability limit of Lieb's [Phys. Rev. Lett. 46, 457 (1981)] bound E{sub 0{>=}}E{sub HF} with unbounded oscillator potentials is given.

Incommensurate ground state of double-layer quantum Hall systems

Double-layer quantum Hall systems possess interlayer phase coherence at sufficiently small layer separations, even without interlayer tunneling. When interlayer tunneling is present, application of a sufficiently strong in-plane magnetic field $B_\parallel > B_c$ drives a commensurate-incommensurate (CI) transition to an incommensurate soliton-lattice (SL) state. We calculate the Hartree-Fock ground-state energy of the SL state for all values of $B_\parallel$ within a gradient approximation, and use it to obtain the anisotropic SL stiffness, the Kosterlitz-Thouless melting temperature for the SL, and the SL magnetization. The in-plane differential magnetic susceptibility diverges as $(B_\parallel - B_c)^{-1}$ when the CI transition is approached from the SL state.

High‐spin behavior of molecular crystals and extended π systems

The qualitative rules for the existence of high-spin ground states in extended systems and molecular crystals are examined here on a firmer theoretical footing. Extended systems have been categorized into three groups, namely, type I, type II, and type III, depending on the type of bonding interactions. The general form of the spin Hamiltonian operators have been written down. The active spaces have been restricted to the minimum size for each of these three types of spin systems. The zeroth-order state vectors and the Hartree–Fock ground-state energies have been identified for unit species of each type. The extended system Hamiltonian operators are further truncated in such a way that only the nearest-neighbor interactions are retained. Expressions have been derived for the energy gap from a molecular orbital approach. The relatively small effects of electron correlation on the energy gaps have been estimated for the type I systems, which belong to the systems of solid-state physics. In particular, it has been shown that for the type I systems the singlet–triplet gap, and hence the ferromagnetic coupling constant, primarily depends upon the difference of one-electron kinetic energies and not on the two-electron exchange integrals. This result agrees with the concept of kinetic exchange that was introduced in the context of a resonating valence-bond formalism. Type II systems are exemplified by extended systems that can be prepared from conjugated molecules while organic molecular crystals form examples of type III species. For these systems, however, the Coulomb exchange interaction has been shown to dominate the energy gap. A quick review of the Heisenberg spin Hamiltonian for the H2 molecule is sufficient to point out that the sign of the calculated ferromagnetic coupling constant depends on the method of calculation, the nature of the basis set, and the bond length. This is amply supported by ab initio calculations on this species. Numerical data have also been obtained from computations on m-phenylene-coupled nitroxy radicals and stacks of α-nitronyl nitroxide, but these calculations have been based on a semiempirical quantum chemical methodology (INDO) since some of the species involved are exceedingly large. Computed energy gaps are in good agreement with experimental and other theoretical (AM1, PM3) results. Nevertheless, for the dimer, trimer, tetramer, and pentamer of the type II specimen, the important π orbitals are far from being degenerate. The quantitative results clearly deviate from the criterion of degeneracy that was suggested from qualitative theories for the existence of a high-spin ground state. Therefore, the criteria for the existence of high spins have been reformulated in terms of the monomer orbitals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 308–324, 2000

Statistics of Hartree-Fock levels in small disordered systems

We study the statistics of quasiparticle and quasihole levels in small interacting disordered systems within the Hartree-Fock approximation. The distribution of the inverse compressibility, given according to Koopmans' theorem by the distance between the two levels across the Fermi energy, evolves from a Wigner distribution in the noninteracting limit to a shifted Gaussian for strong interactions. On the other hand, the nature of the distribution of spacings between neighboring levels on the same side of the Fermi energy (corresponding to energy differences between excited states of the system with one missing or one extra electron) is not affected by the interaction and follows Wigner-Dyson statistics. These results are derived analytically by isolating and solving the appropriate Hartree-Fock equations for the two levels. They are substantiated by numerical simulations of the full set of Hartree-Fock equations for a disordered quantum dot with Coulomb interactions. We find enhanced fluctuations of the inverse compressibility compared to the prediction of the random matrix theory, possibly due to the localization of the wave functions around the edge of the dot. The distribution of the inverse compressibility calculated from the discrete second derivative with respect to the number of particles of the Hartree-Fock ground state energy deviates from the distribution of the level spacing across the Fermi energy. The two distributions have similar shapes but are shifted with respect to each other. The deviation increases with the strength of the interaction thus indicating the breakdown of Koopmans' theorem in the strongly interacting limit.

Accurate universal basis set for H through Xe for Hartree–Fock calculations

We have applied the generator coordinate Hartree–Fock method to generate Slater-type functions for the atoms from H through Xe. The Griffin–Hill–Wheeler–Hartree–Fock equations are integrated numerically generating a universal basis set for these atoms. When compared with the values obtained by Koga et al. [Phys. Rev. A 47 (1993) 4510] using a smaller but fully-optimized basis sets, it is generally observed that our Hartree–Fock ground state energies are equal for the atoms from He ( Z=2) to Mg ( Z=12), larger for the atoms from Al ( Z=13) to Kr ( Z=36), and lower for the atoms from Rb ( Z=37) to Xe ( Z=54). Moreover, our energy results compare satisfactorily with the corresponding atom-optimized and numerical Hartree–Fock calculations of Koga et al. [J. Chem. Phys. 103 (1995) 3000].

A Hartree–Fock ab initio band-structure calculation employing Wannier-type orbitals

An ab initio Wannier-function-based approach to electronic ground-state calculations for crystalline solids is outlined. Using a linear combination of atomic orbitals, the infinite character of the solid is rigorously taken into account. The Hartree–Fock ground-state energy, cohesive energy, lattice constant and bulk modulus are calculated in an ab initio manner, as demonstrated for sodium chloride, using basis sets close to the Hartree–Fock limit. The Hartree–Fock band-structure can be recovered with the current approach and agrees with that obtained from a Bloch-orbital-based calculation. The advantage of the present approach lies in its capability to include electron correlation effects for crystalline insulators by means of quantum chemical procedures.

Distributed Gaussian basis sets: description of the Hartree-Fock ground state energies of N2, CO and BF using s- and p-type Gaussian functions

The ground state Hartree-Fock electronic energy for the nitrogen molecule is determined by using basis sets of distributed s- and p-type Gaussian functions and compared with the results of fully numerical (finite difference) Hartree-Fock calculations. Calculations are also reported for the ground states of the CO and BF molecules and a universal distributed Gaussian basis set established. A potential energy curve is calculated for the CO molecule and the calculation of energy derivatives is considered. The use of distributed basis sets of s and p Gaussian functions in the study of electron correlation effects using many-body perturbation theory is demonstrated.

Hartree-Fock ground-state energy of anyons with no Coulomb interaction in the zero effective field

We find, in the Hartree-Fock approximation, the ground-state energy of anyons with no Coulomb interaction in the case when the external magnetic field precisely cancels the average statistical field. From the point of view of the fractional quantum Hall effect it is shown that statistics transmutations to superfermions at filling fractions v = 1 2 p are not energetically favourable.

STATISTICS TRANSMUTATIONS IN TWO-DIMENSIONAL SYSTEMS AND THE FRACTIONAL QUANTUM HALL EFFECT

Statistics transmutations to superfermions in fractional quantum Hall effect systems are considered in the Hartree-Fock approximation and in the RPA. The Hartree-Fock ground state energy shows that the transmutations are not energetically preferable. Within the RPA it is found that the system exhibits a fractional quantum Hall effect.

Hartree-Fock method posed as a density-functional theory: Application to the Be atom.

The Hartree-Fock ground-state energy and electron density are first shown to be derivable from a local one-body effective potential v(r). As a nontrivial example, attention is then focused on the Be atom and isoelectronic atomic ions, the wave functions being written in terms of the density amplitude and phase. Some related general comments on the two-level one-dimensional system are included; kinetic-energy density is shown to be a local functional of electron density generated by the harmonic-oscillator potential.

Fermionic lattice models with internally competing symmetries: Nontrivial algebraic corrections in the Hartree-Fock ground-state energy.

We show that even for homogeneous ground states the Hartree-Fock (HF) energy of fermionic lattice models may contain nontrivial algebraic corrections in the interaction. Hence the standard HF result for the energy, i.e., the linear dependence on the interaction, is found to be nongeneric. Algebraic corrections will appear in the HF energy whenever the k-space symmetry of the interaction is different from that of the kinetic energy, this being the generic case. The ensuing symmetry conflict leads to a distortion of the noninteracting Fermi surface. The energy corrections are explicitly evaluated for a generalized, two-dimensional Hubbard model with nearest-neighbor hopping and interaction along the diagonal. Algebraic corrections in the HF ground-state energy due to spontaneously broken symmetries may only occur if ${\mathrm{\ensuremath{\epsilon}}}_{\mathbf{k}}$, the dispersion of free particles, has a special form.

Ground-state energy of a two-dimensional charge-density-wave state in a strong magnetic field

The charge-density-wave state of two-dimensional electrons which occupy only the lowest Landau level is investigated. The previous Hartree-Fock calculation by Yoshioka and Fukuyama [J. Phys. Soc. Jpn. 47, 394 (1979)] is extended to include the higher harmonics of the density wave. The correction to the Hartree-Fock ground-state energy is calculated by second-order perturbation theory. Effects of higher Landau levels are also estimated. It is found that the ground-state energy is a smooth function of $\ensuremath{\nu}$, the filling factor of the lowest Landau level.

Dielectric function of the electron gas with dynamical-exchange decoupling. II. Discussion and results

By making the dynamical-exchange decoupling in the equation of motion for the Wigner distribution function, exchange effects in the dielectric function of the homogeneous electron gas were, in an earlier derivation, described by a frequency-dependent local-field correction $G(q,\ensuremath{\omega})$. In paper I, details were provided how the sixfold integral for $G(q,\ensuremath{\omega})$ can be reduced analytically into a double integral, adapted for numerical purposes. In this paper, the consequences of dynamical-exchange effects are studied, and the theory is tested for its internal consistency. The evaluation of the pair correlation function $g(r)$ at the origin from both the static limit and the high-frequency limit of the frequency-dependent local-field correction $G(q,\ensuremath{\omega})$ leads to the same value $g(0)=\frac{1}{2}$, in contrast to other theories where, from both limits, different results are obtained. Also, the compressibility, calculated from the dielectric function including exchange, agrees with the Hartree-Fock result. Furthermore, it is shown that the high-frequency limit of $G(q,\ensuremath{\omega})$ satisfies the general properties implied by the third-frequency-moment sum rule, resulting again in $g(0)=\frac{1}{2}$ and leading to the Hartree-Fock ground-state energy. This consistency between the static and high-frequency behavior of $G(q,\ensuremath{\omega})$ cannot be fulfilled by any static approximation to $G(q,\ensuremath{\omega})$, because an adequate treatment of dynamical-exchange effects involves excited states that are consistent with the Hartree-Fock ground state. In the static limit, $G(q,\ensuremath{\omega})$ exhibits a relatively sharp peak near $q=2{k}_{F}$. This peak induces an instability of the homogeneous electron gas, quite similar to and in the same density range as the instability of the spin susceptibility discussed by Hamann and Overhauser. For ${r}_{s}\ensuremath{\ge}10.6$, a supplementary instability relative to charge-density deformations occurs. The frequency dependence of $G(q,\ensuremath{\omega})$ is examined, and numerical values for $\mathrm{Re}G(q,\ensuremath{\omega})$ and $\mathrm{Im}G(q,\ensuremath{\omega})$ are presented. The real part of the dielectric function and the imaginary part of the inverse dielectric function are plotted for several densities in the metallic range. Compared to the random-phase approximation (RPA), the frequencies of the maxima in the structure factor obtained in the present work (i.e., with dynamical-exchange decoupling) are shifted to lower frequencies. The plasmon dispersion is considerably closer to recent experimental data in aluminium than with RPA. Finally, it turns out that the inclusion of frequency-dependent exchange effects results in the natural occurrence of spin- and charge-density waves. These are the dynamical extension of the instability relative to magnetic perturbations, found in the static limit at low densities.

Atomic correlation energies. III. Second-order corrections to the Hartree-Fock ground state of B, C, N, O and F

For pt.II see ibid., vol.11, no.12, p.2069 (1978). The authors have calculated the second-order perturbative corrections to the Hartree-Fock ground state energies of B, C, N, O and F. The basis used was sufficiently large to assure truncation errors less than 10-3 au in all cases. They found EB(2)=-0.1113 au, EC(2)=-0.1476 au, EN(2)=-0.1840 au, EO(2)=-0.2540 au and EF(2)=-0.3206 au which represent 89.2%, 94.3%, 97.6%, 98.5% and 99.6% respectively of the empirical correlation energies.

Atomic correlation energies. II. Converged E(2)values for Ne and Ne+

For pt.I see ibid., vol.11, p.1157 (1978). The second-order perturbative corrections to the Hartree-Fock ground-state energies of Ne and Ne+ are calculated using a Hartree-Fock-Roothaan approach with practically complete basis sets. The results are ENe(2)=-0.3855 au and ENe+(2)=-0.3214 au. It was found necessary to include virtual orbitals with relatively high angular momenta in the basis sets in order to saturate the E(2) values. The Ne result represents 98.8% of the empirical correlation energy. From ENe(2), ENe+(2) and the Hartree-Fock ground-state energies the authors find an ionisation potential, IPNe,calc=0.7934 au or 21.59 eV, in excellent agreement with the measured value, IPNe=21.56 eV.

Antisymmetry in the three-nucleon-interaction matrix elements

The inclusion of three-body forces to the effective nucleon-nucleon interaction has been frequently applied in nuclear self-consistent Hartree-Fock calculations. Thus, a dependence of the effective force on the nucleon density is exhibited, stimulating great interest in this manner. Antisymmetrized three- body matrix elements are evaluated, and applied with a zero-range three-body force within a finite basis of spherical single-particle states. The imposed antisymmetry ignores the existence of matrix elements of three identical nucleons. It provides for a better understanding of the structure of the three-body contribution to the Hartree-Fock ground state energy and for the exact relation between the three-body force and a density dependent two-body force. The discussion is given as well for spherical as for deformed nuclei. Problems concerning possible overbinding and violation of the spin stability by the zero- range three-body interaction are examined and represented in the angular momentum picture.

An effective exchange potential for the Hartree-fock ground state energy in a solid

A new approximation of the effective exchange potential is derived by a simple method. Its Fourier coefficient V X ( q) is exact when q ⪢ k F (Fermi momentum) (the Kohn and Sham approximation is exact when q ⪢ k F. The discrepancy with a recent result of Kleinman and Langreth is briefly discussed.

Use of the Hartree-Fock Approximation in Computing Electron Affinities

The implications are explored of a recently demonstrated theorem deriving upper bounds on the Hartree-Fock ground-state energy of a system consisting of $N+1$ electrons in the potentials of a finite number of nuclei fixed in a bounded region of space, in terms of energies of single-determinantal wave functions for the system of $N$ electrons in the identical nuclear framework. A conjectured generalization of the theorem is shown to be true. The application of the theorem to a criticism of computations on first-row atoms and ions is discussed, and it is concluded that the use of wave functions which do not satisfy the theorem may be appropriate in the application of the Hartree-Fock approximation to the calculation of electron affinities of certain systems.