Restricted Wave Functions Research Articles

Reciprocal transformation of seniority number restricted wave function.

In this work, an extension of Malmqvist's algorithm is efficiently implemented for the reciprocal transformation of a seniority number restricted multideterminant wave function constructed by nonorthogonal orbitals to an equivalent one constructed by biorthogonal orbitals. Test calculations show that the novel algorithm is capable of applying to transform quite large nonorthogonal orbital based wave function of distributing 22 electrons on 22 orbitals with the maximum seniority number sector and for the evaluation of a three-body operator, at an affordable memory cost of about 800 GB, which is only about one tenth of the size of the corresponding complete active space. Moreover, the algorithm displays improved computational efficiency, due to the compactness of seniority number restricted multi-determinant wave functions.

Composite fermion basis for two-component Bose gases

Despite its success, the composite fermion (CF) construction possesses some mathematical features that have, until recently, not been fully understood. In particular, it is known to produce wave functions that are not necessarily orthogonal, or even linearly independent, after projection to the lowest Landau level. While this is usually not a problem in practice in the quantum Hall regime, we have previously shown that it presents a technical challenge for rotating Bose gases with low angular momentum. These are systems where the CF approach yields surprisingly good approximations to the exact eigenstates of weak short-range interactions, and so solving the problem of linearly dependent wave functions is of interest. It can also be useful for studying higher bands of fermionic quantum Hall states. Here we present several ways of constructing a basis for the space of so-called "simple" CF states for two-component rotating Bose gases in the lowest Landau level, and prove that they all give sets of linearly independent wave functions that span the space. Using this basis, we study the structure of the lowest-lying state using so-called restricted wave functions. We also examine the scaling of the overlap between the exact and CF wave functions at the maximal possible angular momentum for simple states.

Stability of Hemi-Bonded vs Proton-Transferred Structures of (H2O)2+, (H2S)2+, and (H2Se)2+ Studied with Projected Hartree–Fock Methods

Hartree-Fock (HF) is known to suffer from drawbacks in the description of the relative stabilities between the hemi-bonded (HB) and proton-transferred (PT) isomers of the water dimer cation, (H2O)2(+). The energy difference predicted by HF is too large, approximately 27 kcal/mol, which is lowered to 7 kcal/mol when correlation effects are added. The error in HF has been previously attributed to the large dynamic correlation effects in the HB structure as well to the large symmetry breaking this structure exhibits. In this study we use the recently developed projected Hartree-Fock (PHF) methods to study the relative stability of the two isomers of (H2O)2(+) as well as its second and third row analogs, namely, (H2S)2(+) and (H2Se)2(+). In PHF, symmetries are broken and restored in a variation-after-projection approach and thus can deal easily with systems for which HF itself spontaneously breaks symmetry. We use different flavors of PHF (SUHF, KSUHF, SGHF, and KSGHF) to explore their ability in capturing dynamic correlation effects and to compare their performance to different wave function based methods. We study the role of the symmetry-breaking in the above systems, using wave function based methods with unrestricted and restricted wave functions as well as performing a single-shot symmetry restoration (a projection-after-variation scheme).

String Theory on LorentzianAdS3in Minisuperspace

We investigate string theory on Lorentzian AdS_3 in the minisuperspace approximation. The minisuperspace model reduces to the worldline theory of a scalar particle in the Lorentzian AdS_3. The Hilbert space consists of normalizable wave functions, and we see that the unitarity of the theory (or the self-adjointness of the Hamiltonian) restricts the possible sets of wave functions. The restricted wave functions have the property of probability conservation (or current conservation) across the horizons. Two and three point functions are also computed. In the Euclidean model functional forms of these quantities are restricted by the SL(2,R) symmetry almost uniquely, however, in the Lorentzian model there are several ambiguities left. The ambiguities are fixed by the direct computation of overlaps of wave functions.

On the importance of restricted wave function: a comment on the N 2–CO + case

It has been shown that the vibrational frequency calculation of the cationic N 2–CO + radical is not well described at the unrestricted MP2 approach, while the geometrical and energetic properties are rather reliable. The ill-representation of the vibrational frequency disappears when we use the restricted wave function (ROMP2). Even though the ROMP2 method could provide acceptable results, but the use of a more sophisticated method is necessary to obtain results close to the experimental ones.

Density functional calculations of dinuclear organometallic carbonyl complexes. Part I: metalmetal and metalCO bond energies

One of the most fundamental properties in chemistry is the bond dissociation energy (BDE), the energy required to break a specific bond of a molecule. In this paper we apply gradient-corrected density functional theory (DFT) to the calculation of the BDEs of three prototypical organometallic complexes, Mn 2(CO) 10, Fe 2(CO) 9, and Co 2(CO) 8, along with the CO-loss products Mn 2(CO) 9 and Mn 2(CO) 8. We consider the dissociation of both the metalmetal bond and a metalcarbonyl bond. For Mn 2(CO) 10 and Fe 2(CO) 9, the calculated metalmetal BDE is within the error of the experimental measurements. However, the calculated metalmetal BDE for Co 2(CO) 8 is not within the errors of the measurements, but is improved greatly with an unrestricted wavefunction compared with a restricted wavefunction. For the first carbonyl BDE, the calculations agree within a few kcal mol −1 for each complex.

Hole localization versus delocalization in symmetric molecular ions.

Hartree-Fock (HF) and local-density approximation (LDA) calculations of holes in symmetric molecular ions lead to conflicting results on the sign of the energy associated with reducing the symmetry. According to the HF theory, this energy can only be negative because it corresponds to a less restricted wave function. According to the LDA, it can only be positive because it is mostly electrostatic force that concentrates the hole charge in the vicinity of a single atom. In the present paper, we discuss these diverging views and present some LDA results for molecular ions with broken symmetry.

Symmetry breaking in the NO2 σ radical: Construction of the 2A1 and 2B2 states with C s symmetry complete active space self-consistent-field wave functions

Jackels and Davidson’s interpretation of the symmetry-breaking problem in the NO2 radical is examined using complete active space self-consistent-field (CASSCF) wave functions. Jackels and Davidson provided a Lewis-structure analysis of the necessary forms of C2v -restricted wave functions for 2B2 NO2. In particular, C2v and single-configuration restrictions force the molecule to ‘‘choose’’ between the beneficial effects of resonance and the adverse effects of inclusion of charged components in the wave function. This choice, according to Jackels and Davidson, results in a distortion to Cs symmetry in the region of the 2B2 minimum. The Cs symmetry complete active space treatment enables NO2 to achieve wave functions of proper symmetry (for both 2A1 and 2B2 states, when C2v nuclear symmetry is present) while simultaneously yielding significantly localized orbitals, thus offering a means to verify the earlier analysis. Although the Cs complete active space results contradict one important Jackels–Davidson result by predicting a very shallow C2v 2B2 energy minimum for NO2, the authors find the earlier molecular orbital picture of NO2 to be a useful tool for qualitatively understanding its electronic structure.

Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v -representability problem

Universal variational functionals of densities, first-order density matrices, and natural spin-orbitals are explicitly displayed for variational calculations of ground states of interacting electrons in atoms, molecules, and solids. In all cases, the functionals search for constrained minima. In particular, following Percus [Formula: see text] is identified as the universal functional of Hohenberg and Kohn for the sum of the kinetic and electron-electron repulsion energies of an N-representable trial electron density rho. Q[rho] searches all antisymmetric wavefunctions Psi(rho) which yield the fixed. rho. Q[rho] then delivers that expectation value which is a minimum. Similarly, [Formula: see text] is shown to be the universal functional for the electron-electron repulsion energy of an N-representable trial first-order density matrix gamma, where the actual external potential may be nonlocal as well as local. These universal functions do not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus, the v-representability problem, which is especially severe for trial first-order density matrices, has been solved. Universal variational functionals in Hartree-Fock and other restricted wavefunction theories are also presented. Finally, natural spin-orbital functional theory is compared with traditional orbital formulations in density functional theory.

Shape restricted wavefunctions: Ellipsoidal limits for ground and excited σ g states of H 2+

Ellipsoidal as well as spherical limits for the energies of the 1sσ g, 2sσ g and 3sσ g states of H 2 + have been obtained at various internuclear distances. The improvement obtained for the ground-state energy in the ellipsoidal limit is about 40% of the difference between the spherical limit and the exact energy.

Shape restricted wavefunctions: Two center spherical limits for H +2

The two-center spherical limits for the energies, 〈 x 2〉, 〈 z 2〉, equilibrium distances and vibration frequencies are obtained for the ground state of H + 2. It is found, inter alia, that a linear combination of two floating spherical functions approximates the exact ground state energy to within 0.7%.

Study of the Neutron-Proton Correlations in the Shell Model

A shell-model treatment is given for the spectra of the nuclei where there are two protons and two neutrons outside a doubly closed shell. The only restriction is that like-nucleons are required to occupy the same angular momentum state. In addition to this general wave function, more restricted wave functions also are investigated, viz., the pairing wave function (as least one pair of like nucleons is coupled to zero angular momentum) which contain no $n\ensuremath{-}p$ correlations; the quartetting wave function (two unlike nucleons are first coupled to the maximum angular momentum, then the two pairs are coupled to $I$, the nuclear angular momentum) which does contain $n\ensuremath{-}p$ correlations; and a superposition of pairing and quartetting wave functions. Results for $^{44}\mathrm{Ti}$, $^{64}\mathrm{Zn}$, and $^{204}\mathrm{Hg}$ are given. The ground states are reproduced quite well by the pairing wave function; it seems, however, to fail for the excited states. The overlap between the general and the quartetting wave function is quite large. Finally, results are also given for the pairing force, which has only a $T=1$ component.

Quantum Mechanics of the H2–H2 Interaction. III. Nonorthogonal SCF—GF Calculations in the One-Configuration Approximation

Short-range intermolecular forces in the H2–H2 system have been approached by a rigorous quantum-mechanical calculation using a wavefunction consisting of one configuration of nonorthogonal group functions constructed from a minimal basis set of 1s Slater orbitals. The need of avoiding, with restricted wavefunctions, the forced orthogonalization of the atomic-orbital basis, which amounts to the relaxation of the strong orthogonality condition in a group-function approach, appears to be essential in order to give the correct angular dependence of the intermolecular energy. About 98% of the potential `barrier' resulting from the minimal-basis-set calculation with complete configuration interaction is accounted for by nonorthogonal group functions which allow only for intramolecular correlation. The even simpler one-determinant wavefunction constructed from SCF bond orbitals gives numerical results within 5% to 10% of the reference value for the intersection and within 3% to 5% for the energy difference between different relative orientations. Both inter- and intramolecular correlations seem therefore to be relatively unimportant in determining the relative orientation of the two molecules in the short-range region. The partitioning of the short-range interaction energy into a Coulomb component (slightly attractive and slowly varying with the dihedral angle) and into a correction or penetration term, which arises from electron interchange between the electron groups when they begin to overlap, shows that the largest part of the interaction and its orientation dependence arise from the repulsion due to negative overlap between the closed-shell charge clouds.

Hartree-Fock Theory: Slater Determinants of Minimum Energy

A theorem is proved (on the basis of physically reasonable assumptions) to the effect that in Hartree-Fock (HF) theory an electron can always be added to an $N$-electron system without raising its energy. The theorem is applied to show that the conventional HF wave function is unstable for certain negative atomic ions (${\mathrm{H}}^{\ensuremath{-}}$, ${\mathrm{Li}}^{\ensuremath{-}}$ ${\mathrm{B}}^{\ensuremath{-}}$, ${\mathrm{N}}^{\ensuremath{-}}$, ${\mathrm{N}}^{\mathrm{---}}$, ${\mathrm{O}}^{\ensuremath{-}}$, ${\mathrm{O}}^{\mathrm{---}}$, ${\mathrm{Na}}^{\ensuremath{-}}$, ${\mathrm{P}}^{\ensuremath{-}}$), and that the typical extent of the instability is chemically significant. A consequence of the instability is that the lowest level in HF theory does not have the symmetry of the exact ground level. This is the case, in particular, for the closed-shell ions ${\mathrm{H}}^{\ensuremath{-}}$, ${\mathrm{Li}}^{\ensuremath{-}}$, ${\mathrm{O}}^{\mathrm{---}}$, and ${\mathrm{Na}}^{\ensuremath{-}}$---a result contrary to some earlier expectations. The two-electron one-center system with fixed nuclear charge $\mathrm{Ze}$ ($Z$ not restricted to integer values) is investigated in detail on the basis of a simplified model defined by specifying restrictions which the allowable determinantal wave functions must satisfy. The orbitals of the determinantal wave functions of the model are of the form ${\ensuremath{\psi}}_{i}={v}_{i}{\ensuremath{\chi}}_{i}$, $i=1,2$, where ${\ensuremath{\chi}}_{i}$ is a spin function and ${v}_{i}$ is essentially of $1s$ hydrogenic form. As $Z$ decreases, the model shows explicitly the onset of the instability described above. It further shows that for $Zg1$, any true minimum-energy determinant ${D}_{g}$ is bound (although the interesting question of whether ${D}_{g}$ is bound for $Zl~1$, apparently raised here for the first time, remains unanswered). A precise analogy between the electron problem and certain spin problems is defined. This analogy sheds light on the HF approximation. It leads to examples which show strikingly that it is important not to always impose symmetry restrictions on an otherwise restricted wave function; the analogy also makes clear, in a simple way, certain points fundamental to an understanding of HF theory. Some possible implications of the results of this paper for solid-state calculations are discussed.