Hartree-Fock Ground State Research Articles

A variationally stable compact Hartree–Fock-style wavefunction for a non-degenerate first excited state

A compact approximate wavefunction for a non-degenerate first excited state constructed as a Slater determinant from which the ground state Hartree–Fock (HF) wavefunction has been projected out explicitly is introduced, together with all expressions required for its calculation through gradient-based energy optimization schemes. The explicit projection makes this wavefunction suitable for the description of a non-degenerate first excited state which is of the same symmetry as the HF ground state without any danger of a variational collapse to the lower energy ground state. The equations allowing the calculation of a spin-unrestricted version of the proposed wavefunction have been implemented in code. The results of excited state calculations on HeH+, H2, C2, N2, H2O and H2O2 show that this excited state wavefunction which, effectively, is comprised of just two Slater determinants, only one of which is optimized variationally, is capable of producing results which compare favourably to those coming from standard approaches such as single-excitation configuration interaction (CIS), time-dependent HF (TDHF), symmetry-adapted-cluster CI (SAC-CI) and equations-of-motion coupled-cluster with singles and doubles (EOM-CCSD).

Hartree–Fock ground-state properties for the group 1 alkali metals and the group 11 noblemetals

In order to use wavefunction-based correlation methods in solids it is necessary to havereliable Hartree–Fock results for the infinite system of interest. Therefore weperformed Hartree–Fock calculations for the group 1 alkali metals (Li to Cs)and group 11 noble metals (Cu, Ag and Au). We optimized a basis set ofvalence-double-ζ quality for the periodic system. For the lighter atoms all-electron basis sets are applied,whereas for the heavier atoms small-core pseudopotentials with the corresponding basis setswere used to deal with the scalar-relativistic effects. We determine the cohesive energy, thelattice constant and the bulk modulus of the systems at the Hartree–Fock level. We use thecounterpoise correction for the free atom to minimize the basis set superposition erroroccurring for finite basis sets. The effects due to the counterpoise correction not only forthe cohesive energy but also for the lattice structure and bulk modulus are discussed indetail.

Correlation energies in distorted 3d-t 2g perovskite oxides

Using an effective low-energy Hamiltonian derived from the first-principles electronic structure calculations for the narrow t2g bands of YTiO3, LaTiO3, YVO3, and LaVO3, we evaluate the contributions of the correlation energy Ec to the stability of different magnetic structures that can be realized in these distorted perovskite oxides. We consider two approximations for Ec that are based on regular perturbation theory expansion around a nondegenerate Hartree-Fock ground state. One is the second order of perturbation theory, which allows comparing the effects of local and nonlocal correlations. The other is the local t-matrix approach, which allows treating some higher-order contributions to Ec. The correlation effects systematically improve the agreement with the experimental data and additionally stabilize the experimentally observed G- and C-type antifer-romagnetic (AFM) structures in YVO3 and LaVO3, although the absolute magnitude of the stabilization energy is sensitive to the level of approximations and is somewhat smaller in the t-matrix method. The nonlocal correlations additionally stabilize the ferromagnetic ground state in YTiO3 and the C-type AFM ground state in LaVO3. Among two inequivalent transition-metal sites in the monoclinic structure, the local correlations are stronger at the sites with the least distorted environment. Limitations of the regular perturbation-theory expansion for LaTiO3 are also discussed.

Calculation of K-edge circular dichroism of amino acids: Comparison of random phase approximation with other methods

Soft x-ray natural circular dichroism of amino acids is studied by means of ab initio methods. Several approaches to evaluate the oscillator and rotary strengths of core-to-valence excitations are compared from the viewpoint of basis set dependence: ground-state Hartree-Fock (HF) orbital set employed in (i) random phase approximation (RPA), (ii) static exchange approach (STEX) (unrelaxed), (iii) core-ionized state HF orbital set applied in STEX (relaxed), and (iv) HF excited state orbital set for each core-to-valence excited state. Furthermore in (i) the PRA in the framework of the density functional method (DFT) is compared with the RPA where the ab initio HF orbital set is used. In (iv), the oscillator and rotary strengths evaluated by different orbital sets for the initial and final states, namely, nonorthogonal ground-state and core-excited HF orbitals, are compared with those evaluated by using the core-excited HF orbital set to describe the initial (ground) state. It was shown that, among considered methods, the RPA provides most consistent and less time-consuming results for circular dichroism core excitation spectra. Discussion of the low energy part of K edge circular dichroism spectra of five common amino acids obtained with the help of RPA is presented.

Nuclear collective excitations using correlated realistic interactions: The role of explicit random-phase approximation correlations

We examine to what extent correlated realistic nucleon-nucleon interactions, derived within the unitary correlation operator method (UCOM), can describe nuclear collective motion in the framework of first-order random-phase approximation (RPA). To this end we employ the correlated Argonne V18 interaction in calculations within the so-called ''Extended'' RPA (ERPA) and investigate the response of closed-shell nuclei. The ERPA is a renormalized RPA version which considers explicitly the depletion of the Fermi sea due to long-range correlations and thus allows us to examine how these affect the excitation spectra. It is found that the effect on the properties of giant resonances is rather small. Compared to the standard RPA, where excitations are built on top of the uncorrelated Hartree-Fock (HF) ground state, their centroid energies decrease by up to 1 MeV, approximately, in the isovector channel. The isoscalar response is less affected in general. Thus, the disagreement between our previous UCOM-based RPA calculations and the experimental data are to be attributed to other effects, namely to residual three-body terms in the Hamiltonian and to higher-order configurations. Single-particle energies and occupation probabilities obtained within the ERPA are compared with corresponding HF and perturbation-theory results and are discussed as well. The ERPA formalism ismore » presented in detail.« less

Exchange force for two-level systems such as LiH and [formula omitted

Using the Dawson–March transformation, the Dirac density matrix γ( r, r′) can be written in terms of the density amplitude ρ ( r ) and a phase θ( r), where ρ( r) ≡ γ( r, r) is the ground-state electron density. Here, for systems such as LiH or H 3 - , it is shown how the force −∂ V( r)/∂ r corresponding to the one-body potential V( r) can be written, given the ground-state density ρ( r) from a high-level ab initio calculation or a diffraction experiment. The Hartree–Fock ground-state densities for LiH and H 3 - are utilized to calculate approximate phase angles θ( r). Finally, for H 3 - the force −∂ V( r)/∂ r corresponding to V( r) is calculated.

Collective multipole excitations based on correlated realistic nucleon-nucleon interactions

We investigate collective multipole excitations for closed-shell nuclei from $^{16}\mathrm{O}$ to $^{208}\mathrm{Pb}$ by using correlated realistic nucleon-nucleon interactions in the framework of the random phase approximation (RPA). The dominant short-range central and tensor correlations are treated explicitly within the unitary correlation operator method (UCOM), which provides a phase-shift-equivalent correlated interaction ${V}_{\mathrm{UCOM}}$ adapted to simple uncorrelated Hilbert spaces. The same unitary transformation that defines the correlated interaction is used to derive correlated transition operators. Using ${V}_{\mathrm{UCOM}}$, we solve the Hartree-Fock (HF) problem and employ the single-particle states as a starting point for the RPA. By construction, the UCOM-RPA is fully self-consistent, i.e., the same correlated nucleon-nucleon interaction is used in calculations of the HF ground state and in the residual RPA interaction. Consequently, the spurious state associated with the center-of-mass motion is properly removed, and the sum rules are exhausted within $\ifmmode\pm\else\textpm\fi{}3%$. The UCOM-RPA scheme results in a collective character of giant monopole, dipole, and quadrupole resonances in closed-shell nuclei across the nuclear chart. For the isoscalar giant monopole resonance, the resonance energies are in agreement with experiment, hinting at a reasonable compressibility. However, in the ${1}^{\ensuremath{-}}$ and ${2}^{+}$ channels the resonance energies are overestimated because of missing long-range correlations and three-body contributions.

Imaging molecular orbitals using photoionization

The interpretation of a recent experiment using high-order harmonic generation [Itatani et al., Nature 432 (2004) 867] as a measurement of the highest occupied molecular orbital of a molecule is conceptually problematic, even if the independent-particle picture is taken seriously. Guided by the relationship between the amplitude for one-photon-induced electron emission and the electron–ion recombination amplitude in the three-step model of high-order harmonic generation, it is argued that synchrotron-based photoionization might be a superior approach to imaging molecular orbitals. Within the Hartree–Fock independent-particle picture, the molecular-frame photoelectron angular distributions, measured as a function of photon energy, could be used to reconstruct all orbitals occupied in the Hartree–Fock ground state of the molecule investigated. It is suggested that laser alignment techniques could be employed to facilitate the measurement of the molecular-frame photoelectron angular distributions.

Optimized effective potentials yielding Hartree–Fock energies and densities

It is commonly believed that the exchange-only optimized effective potential (OEP) method must yield total energies that are above corresponding ground-state Hartree-Fock (HF) energies except for one- and two-electron systems. We present a simple procedure for constructing local (multiplicative) exchange potentials that reproduce exactly the HF energy and density in any finite basis set for any number of electrons. For any finite basis set, no matter how large, there exist infinitely many such OEPs, which questions their suitability for practical applications.

Proposal for a modified Møller-Plesset perturbation theory

A modified version of the Moller-Plesset approach for obtaining the correlation energy associated to a Hartree-Fock ground state is proposed. The method is tested in a model of interacting fermions that allows for an exact solution. Using up to third order terms improved results are obtained, even more accurate in the limit of loosely bound particles. This result suggests the possible convenience of the scheme for the study of chemical bound problems.

Proposal for a modified Moeller-Plesset perturbation theory

A modified version of the Moeller-Plesset approach for obtaining the correlation energy associated with a Hartree-Fock ground state is proposed. The method is tested in a model of interacting fermions that allows for an exact solution. Using up to third order terms improved results are obtained, even in the limit of loosely bound particles. Tested in molecules as well, the modified method appears to give improved results in symmetric systems.

Gaussian Basis Sets for Some Polyatomic Two‐Heavy‐Atom Hydrides

AbstractFor Abstract see ChemInform Abstract in Full Text.

Variational grand-canonical electronic structure method for open systems

An ab initio method is developed for variational grand-canonical molecular electronic structure of open systems based on the Gibbs-Peierls-Boguliobov inequality. We describe the theory and a practical method for performing the calculations within standard quantum chemistry codes using Gaussian basis sets. The computational effort scales similarly to the ground-state Hartree-Fock method. The quality of the approximation is studied on a hydrogen molecule by comparing to the exact Gibbs free energy, computed using full configuration-interaction calculations. We find the approximation quite accurate, with errors similar to those of the Hartree-Fock method for ground-state (zero-temperature) calculations. A further demonstration is given of the temperature effects on the bending potential curve for water. Some future directions and applications of the method are discussed. Several appendices give the mathematical and algorithmic details of the method.

Collective excitations in the unitary correlation operator method and relativistic QRPA studies of exotic nuclei

The collective excitation phenomena in atomic nuclei are studied in two different formulations of the Random Phase Approximation (RPA): (i) RPA based on correlated realistic nucleon-nucleon interactions constructed within the Unitary Correlation Operator Method (UCOM), and (ii) relativistic RPA (RRPA) derived from effective Lagrangians with density-dependent meson-exchange interactions. The former includes the dominant interaction-induced short-range central and tensor correlations by means of an unitary transformation. It is shown that UCOM-RPA correlations induced by collective nuclear vibrations recover a part of the residual long-range correlations that are not explicitly included in the UCOM Hartree-Fock ground state. Both RPA models are employed in studies of the isoscalar monopole resonance (ISGMR) in closed-shell nuclei across the nuclide chart, with an emphasis on the sensitivity of its properties on the constraints for the range of the UCOM correlation functions. Within the Relativistic Quasiparticle RPA (RQRPA) based on Relativistic Hartree-Bogoliubov model, the occurrence of pronounced low-lying dipole excitations is predicted in nuclei towards the proton drip-line. From the analysis of the transition densities and the structure of the RQRPA amplitudes, it is shown that these states correspond to the proton pygmy dipole resonance.

Theoretical studies of ground and excited electronic states of OLED material bis(2-methyl-8-quinolinolato)gallium(iii) chlorine

By means of ab initio HF and DFT B3LYP methods, the structure of bis(2-methyl-8-quinolinolato)gallium(III) chlorine complex(GaMq 2Cl) was optimized and the electronic transition mechanism was studied in the complex . The lowest singlet excited state (S1) of GaMq 2Cl has been studied by the singles configuration interaction (CIS) method and time-dependent density functional theory (TD-DFT). The lowest singlet electronic transition (S0→S1) of GaMq 2Cl is π–π* electronic transitions and primarily localized on the phenol and pyridyl ligands. The emission of GaMq 2Cl is due to the electron transitions from the phenol donor to the pyridyl acceptor including C→C and O→N transference. Two possible electron transfer pathways are presented, one by carbon, oxygen and nitrogen atoms, and the other via metal cation Ga 3+. The comparison between the CIS optimized excited-state structure and the Hartree-Fock ground-state structure indicates that the geometric shift is mainly confined to the one quinoline and these changes can be easily understood in terms of the nodal patterns of the highest occupied and lowest unoccupied molecular orbitals. TD-B3-LYP calculations predict an emission wavelength of 504.57 nm. This is comparable to GaMq 2Cl 492 nm observed experimentally for photoluminescence. Lending theoretical corroboration to recent experimental observations and supposition, the nature of the electron transition mechanism was revealed.

GSA algorithm applied to electronic structure: Hartree–Fock–GSA method

AbstractA stochastical algorithm to investigate the real closed‐shell Hartree–Fock problem is proposed. The approach is based on a global optimization method, the generalized simulated annealing. We tested this methodology by determining the Hartree–Fock ground state of the H2, LiH, BH, Li2, OH−, FH, CO, N2, BeH2, CH2, H2O, NH3, HCHO, CH4 molecular systems using minimal, double‐zeta, and triple‐zeta bases. The main characteristic of this method is that it enables mapping the electronic hypersurface to find minima with the guarantee of finding the absolute minimum of the functional in focus. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Extended RPA with ground-state correlations in a solvable model

The ground states and excited states of the Lipkin model hamiltonian are calculated using a new theoretical approach which has been derived from an extended time-dependent Hartree-Fock theory known as the time-dependent density-matrix theory (TDDM). TDDM enables us to calculate correlated ground states, and its small amplitude limit (STDDM), which is a version of extended RPA theories based on a correlated ground state, can be used to calculate excited states. It is found that this TDDM plus STDDM approach gives much better results for both the ground states and the excited states than the Hartree-Fock ground state plus RPA approach.

The two-level Lipkin model solution within the RPA approach for deformed fermions

We use a deformed algebra formalism, the so-called quon algebra, to investigate the role of deformation in an approximate treatment of a many-body system in a simple schematic model. A previously developed method is used in order to find the Hartree–Fock ground state energy and the RPA frequency as a function of the deformation parameter. We show that this dependence is very sensitive to small deviations of the algebra from the pure fermionic case and that the frequency is renormalized by those deviations. Also, we conclude that the well-known RPA collapse still occurs, although the critical value for the interaction constant now depends on the deformation parameter. Finally, we show that it is possible to make our results converge to the exact solution, allowing for a dependence of the deformation parameter on the two-body interaction strength of the model.

Distributed Gaussian basis sets: Variationally optimized s‐type sets for the open‐shell systems HeH and BeH

AbstractDistributed basis sets of s‐type Gaussian functions are determined by invoking the variation principle for the restricted open‐shell matrix Hartree–Fock ground states of the open‐shell molecular systems HeH and BeH for nuclear separations of 1.500 bohr and 2.500 bohr, respectively. The calculated energy expectation values supported by these distributed basis sets are compared with the energies obtained from finite difference open‐shell Hartree–Fock calculations. The restricted open‐shell matrix Hartree–Fock calculations are performed by means of the asymptotic method. The accuracy of the approximations to the energy expectation values supported by the distributed basis sets of s‐type Gaussian functions is comparable with that attained in previous studies of closed‐shell systems. The parameters, that is, the exponents and positions defining the variationally optimized distributed basis sets, are presented and discussed. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr

We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coefficients and orbital exponents are fully optimized in atomic Hartree–Fock (HF) calculations. As opposed to other quadruple zeta basis sets, the basis set errors in atomic ground-state HF energies are less than 1 mEh and increase smoothly across the Periodic Table, while the number of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in atomic MP2 calculations, and for a few cases determined at the HF level for excited atomic states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and density functional theory (DFT) calculations, and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in molecular HF, DFT, and MP2 calculations for a sample of diatomic and small polyatomic molecules by a comparison of energies, bond lengths, and dipole moments with results obtained numerically or using very large basis sets. It is shown that basis sets of quadruple zeta quality are necessary to achieve an accuracy of 1 kcal/mol per bond in HF and DFT atomization energies. For compounds containing third row as well as alkaline and earth alkaline metals it is demonstrated that the inclusion of high-lying core orbitals in the active space can be necessary for accurate correlated treatments. The QZVPP basis sets provide sufficient flexibility to polarize the core in those cases. All test calculations indicate that the new basis sets lead to consistent accuracies in HF, DFT, or correlated treatments even in critical cases where other basis sets may show deficiencies.