A consistent set of ab initio effective core potentials (ECP) has been generated for the main group elements from Na to Bi using the procedure originally developed by Kahn. The ECP’s are derived from all‐electron numerical Hartree–Fock atomic wave functions and fit to analytical representations for use in molecular calculations. For Rb to Bi the ECP’s are generated from the relativistic Hartree–Fock atomic wave functions of Cowan which incorporate the Darwin and mass–velocity terms. Energy‐optimized valence basis sets of (3s3p) primitive Gaussians are presented for use with the ECP’s. Comparisons between all‐electron and valence‐electron ECP calculations are presented for NaF, NaCl, Cl2, Cl2−, Br2, Br2−, and Xe2+. The results show that the average errors introduced by the ECP’s are generally only a few percent.

Ab initio effective core potentials for molecular calculations. Potentials for main group elements Na to Bi

The realization of superfluidity in a dilute gas of fermionic atoms, analogous to superconductivity in metals, represents a long-standing goal of ultracold gas research. In such a fermionic superfluid, it should be possible to adjust the interaction strength and tune the system continuously between two limits: a Bardeen–Cooper–Schrieffer (BCS)-type superfluid (involving correlated atom pairs in momentum space) and a Bose–Einstein condensate (BEC), in which spatially local pairs of atoms are bound together. This crossover between BCS-type superfluidity and the BEC limit has long been of theoretical interest, motivated in part by the discovery of high-temperature superconductors1,2,3,4,5,6,7,8,9,10. In atomic Fermi gas experiments superfluidity has not yet been demonstrated; however, long-lived molecules consisting of locally paired fermions have been reversibly created11,12,13,14,13. Here we report the direct observation of a molecular Bose–Einstein condensate created solely by adjusting the interaction strength in an ultracold Fermi gas of atoms. This state of matter represents one extreme of the predicted BCS–BEC continuum.

https://zenodo.org/record/1233267/files/article.pdf

Emergence of a molecular Bose–Einstein condensate from a Fermi gas

The ground‐state energies of H2, LiH, Li2, and H2O are calculated by a fixed‐node quantum Monte Carlo method, which is presented in detail. For each molecule, relatively simple trial wave functions ΨT are chosen. Each ΨT consists of a single Slater determinant of molecular orbitals multiplied by a product of pair‐correlation (Jastrow) functions. These wave functions are used as importance functions in a stochastic approach that solves the Schrodinger equation by treating it as a diffusion equation. In this approach, ΨT serves as a ‘‘guiding function’’ for a random walk of the electrons through configuration space. In the fixed‐node approximation used here, the diffusion process is confined to connected regions of space, bounded by the nodes (zeros) of ΨT. This approximation simplifies the treatment of Fermi statistics, since within each region an electronic probability amplitude is obtained which does not change sign. Within these approximate boundaries, however, the Fermi problem is solved exactly. The e...

/pdf/fixed-node-quantum-monte-carlo-for-molecules-q4qjld56cm.pdf

Fixed-node quantum Monte Carlo for molecules

The realization of superfluidity in a dilute gas of fermionic atoms, analogous to superconductivity in metals, represents a long-standing goal of ultracold gas research. In such a fermionic superfluid, it should be possible to adjust the interaction strength and tune the system continuously between two limits: a Bardeen–Cooper–Schrieffer (BCS)-type superfluid (involving correlated atom pairs in momentum space) and a Bose–Einstein condensate (BEC), in which spatially local pairs of atoms are bound together. This crossover between BCS-type superfluidity and the BEC limit has long been of theoretical interest, motivated in part by the discovery of high-temperature superconductors. In atomic Fermi gas experiments superfluidity has not yet been demonstrated; however, long-lived molecules consisting of locally paired fermions have been reversibly created. Here we report the direct observation of a molecular Bose–Einstein condensate created solely by adjusting the interaction strength in an ultracold Fermi gas of atoms. This state of matter represents one extreme of the predicted BCS–BEC continuum.

Emergence of a molecular Bose-Einstein condensate from a Fermi gas

Following the realization of Bose–Einstein condensates in atomic gases, an experimental challenge is the production of molecular gases in the quantum regime. A promising approach is to create the molecular gas directly from an ultracold atomic gas; for example, bosonic atoms in a Bose-Einstein condensate have been coupled to electronic ground-state molecules through photoassociation1 or a magnetic field Feshbach resonance2. The availability of atomic Fermi gases offers the prospect of coupling fermionic atoms to bosonic molecules, thus altering the quantum statistics of the system. Such a coupling would be closely related to the pairing mechanism in a fermionic superfluid, predicted to occur near a Feshbach resonance3,4. Here we report the creation and quantitative characterization of ultracold 40K2 molecules. Starting with a quantum degenerate Fermi gas of atoms at a temperature of less than 150 nK, we scan the system over a Feshbach resonance to create adiabatically more than 250,000 trapped molecules; these can be converted back to atoms by reversing the scan. The small binding energy of the molecules is controlled by detuning the magnetic field away from the Feshbach resonance, and can be varied over a wide range. We directly detect these weakly bound molecules through their radio-frequency photodissociation spectra; these probe the molecular wavefunction, and yield binding energies that are consistent with theory.

/pdf/creation-of-ultracold-molecules-from-a-fermi-gas-of-atoms-syjrsftj55.pdf

Creation of ultracold molecules from a Fermi gas of atoms

The multiconfiguration self‐consistent field wave functions and potential energy curves for the lowest lying 1Σ+g, 1Σ+u, 3Σ+g, 3Σ+u, 1Πg, 1Πu, 3Πg, and 3Πu states of Na2 are reported. We find good agreement with recent experimental estimates of the dissociation energies for the 1Σ+g and 1Σ+u states. We find the long range hump in the B 1Πu curve to have its maximum height of 520 cm−1 above the separated atom asymptote at R=6.45±0.1 A; recent experimental estimates of the height are 474 and 554±120 cm−1.

The molecular electronic structure of the lowest 1Σ+g, 3Σ+u, 1Σ+u, 3Σ+g, 1Πu, 1Πg, 3Πu, and 3Πg states of Na2

The molecular electronic structure of the twenty-six lowest lying states of Li2 at short and intermediate internuclear separations

The potential energy curves for the X 1Σ+g and the A 1Σ+u states of Li2 have been calculated on the single configuration Hartree–Fock–Roothaan (HF) level, and on the multiconfiguration self‐consistent‐field (MCSCF) level. The MCSCF results give binding energies De (X 1Σ+g) =8297 cm−1 and De (A 1Σ+u) =9299 cm−1; a semiempirical scaling which reproduces the experimental vibrational energy level spacings suggests ’’most‐likely’’ dissociation energies De′ (X 1Σ+g=8450±100 cm−1 and De′ (A  (1Σ+u) =9400±100 cm−1.

The electronic structure and spectra of the X 1Σ+g and A 1Σ+u states of Li2

We calculate the potential energy curves for all molecular states of NaK which may be obtained from the interactions Na(3s)+K(4s), Na(3s)+K(4p), Na(3p)+K(4s), Na(3s)+K(5s), Na(3s)+K(3d), and for the Δ states corresponding to the interactions Na(3d)+K(4s) and Na(4p)+K(4p) by full‐valence configuration interaction computations which utilize effective core potentials to describe the core electrons, the core‐valence orthogonality constraints, and the core‐valence correlation (CVC) energy. The differences between our computed curves and those deduced from experimental spectra are generally small and can be accounted for by: (1) the modest size of the basis set, which is insufficiently diffuse to describe Na− and K− resonances associated with excited charge transfer interactions and related molecular Rydberg character, and (2) the approximate way in which the CVC interaction is included.

Daniel D. Konowalow

Papers

The molecular electronic structure of the lowest 1Σ+g, 3Σ+u, 1Σ+u, 3Σ+g, 1Πu, 1Πg, 3Πu, and 3Πg states of Na2

The molecular electronic structure of the twenty-six lowest lying states of Li2 at short and intermediate internuclear separations

The electronic structure and spectra of the X 1Σ+g and A 1Σ+u states of Li2

Electronic structure and spectra of the lowest five 1Σ+ and 3Σ+ states, and lowest three 1Π, 3Π, 1Δ, and 3Δ states of NaK

Accurate potential energy curves for the 3Σ+u and b3Σ+g states of Li2