Ultracold molecules provide opportunities for exploring quantum matter, chemical dynamics and information processing thanks to their rich interactions, which can be controlled by external fields. Magnetic fields tune interactions through Feshbach resonances, enabling the formation of ultracold dimers and triatomic molecules from atom-dimer collisions. Here we demonstrate electric-field control of atom-molecule Feshbach resonances. In mixtures of ground-state sodium-potassium molecules and potassium atoms, electric fields shift resonance positions systematically, revealing specific trimer bound states and their electric-field dependent energies. The response differs markedly from isolated dimers, showing hindered rotation of the molecular constituent near an atom. Electric fields therefore add an independent knob for atom-molecule resonances, open spectroscopic access to triatomic quantum states, and advance controlled polyatomic quantum matter.
We report the creation of ultracold bosonic $^{23}$Na$^{87}$Rb Feshbach molecules via magneto-association. By ramping the magnetic field across an interspecies Feshbach resonance, at least 4000 molecules can be produced out of the near degenerate ultracold mixture. Fast loss due to inelastic atom-molecule collisions is observed, which limits the pure molecule number, after residual atoms removal, to 1700. The pure molecule sample can live for 21.8(8) ms in the optical trap, long enough for future molecular spectroscopy studies toward coherently transferring to the singlet ro-vibrational ground…
We reveal the existence of high-density Feshbach resonances in the collision between the ground and metastable states of $^{171}$Yb and coherently produce the associated Feshbach molecules by photoassociation. The extremely small transition rate is overcome by the enhanced Franck-Condon factor of the weakly bound Feshbach molecule, allowing us to observe Rabi oscillations with long decay time between an atom pair and a molecule in an optical lattice. We also perform the precision measurement of the binding energies, which characterizes the observed resonances. The ultra-narrow photoassociation…
We have realized long-lived ground-state polar molecules in a 3D optical lattice, with a lifetime of up to 25 s, which is limited only by off-resonant scattering of the trapping light. Starting from a 2D optical lattice, we observe that the lifetime increases dramatically as a small lattice potential is added along the tube-shaped lattice traps. The 3D optical lattice also dramatically increases the lifetime for weakly bound Feshbach molecules. For a pure gas of Feshbach molecules, we observe a lifetime of >20 s in a 3D optical lattice; this represents a 100-fold improvement over previous resu…
We describe the formation of fermionic NaLi Feshbach molecules from an ultracold mixture of bosonic 23Na and fermionic 6Li. Precise magnetic field sweeps across a narrow Feshbach resonance at 745 G result in a molecule conversion fraction of 5% for our experimental densities and temperatures, corresponding to a molecule number of 5*10^4. The observed molecular decay ifetime is 1.3 ms after removing free Li and Na atoms from the trap.
We demonstrate the transfer of $^{23}$Na$^{40}$K molecules from a closed-channel dominated Feshbach-molecule state to the absolute ground state. The Feshbach molecules are initially created from a gas of sodium and potassium atoms via adiabatic ramping over a Feshbach resonance at 78.3$\,$G. The molecules are then transferred to the absolute ground state using stimulated Raman adiabatic passage with an intermediate state in the spin-orbit-coupled complex $|c^3 Σ^+, v=35, J=1 \rangle \sim |B^1Π, v=12, J=1\rangle$. Our measurements show that the pump transition dipole moment linearly increases w…
We report on the formation of ultracold fermionic Feshbach molecules of $^{23}$Na$^{40}$K, the first fermionic molecule that is chemically stable in its ground state. The lifetime of the nearly degenerate molecular gas exceeds 100 ms in the vicinity of the Feshbach resonance. The measured dependence of the molecular binding energy on the magnetic field demonstrates the open-channel character of the molecules over a wide field range and implies significant singlet admixture. This will enable efficient transfer into the singlet vibrational ground state, resulting in a stable molecular Fermi gas …
We present the prospects for photoassociation, optical control of interspecies scattering lengths and finally, the production of ultracold absolute ground state molecules in the Rb+Hg system. We use the "gold standard" ab initio methods for the calculations of ground (CCSD(T)) and excited state (EOM-CCSD) potential curves. The RbHg system, thanks to the wide range of stable Hg bosonic isotopes, offers possibilities for mass-tuning of ground state interactions. The optical lengths describing the strengths of optical Feshbach resonances near the Rb transitions are favorable even at large laser d…
We show that the probability of the collision-induced absorption of non-resonant microwave photons by a gas of ultracold molecules is dramatically enhanced near a Feshbach scattering resonance. This can be used for detecting Feshbach resonances of ultracold molecules by measuring the microwave field absorption and for tuning the elastic scattering cross sections of ultracold molecules by varying the frequency and intensity of the microwave field in a wide range of the field parameters.
We selectively create p-wave Feshbach molecules in the $m_{l}=\pm 1$ orbital angular momentum projection state of $^{6}$Li. We use an optical lattice potential to restrict the relative momentum of the atoms such that only the $m_{l}=\pm 1$ molecular state couples to the atoms at the Feshbach resonance. We observe the hollow-centered dissociation profile, which is a clear indication of the selective creation of p-wave molecules in the $m_{l}=\pm1$ states. We also measure the dissociation energy of the p-wave molecules created in the optical lattice and develop a theoretical formulation to expla…
Resonances in ultracold collisions involving heavy molecules are difficult to understand, and have proven challenging to detect. Here we report the observation of magnetically tunable Feshbach resonances in ultracold collisions between $^{23}$Na$^{40}$K molecules in the rovibrational ground state and $^{40}$K atoms. We prepare the atoms and molecules in various hyperfine levels of their ground states and observe the loss of molecules as a function of the magnetic field. The atom-molecule Feshbach resonances are identified by observing an enhancement of the loss rate coefficients. We have obser…
We present investigations of the formation rate and collisional stability of lithium Feshbach molecules in an ultracold three-component mixture composed of two resonantly interacting fermionic 6-Li spin states and bosonic 174-Yb. We observe long molecule lifetimes (> 100 ms) even in the presence of a large ytterbium bath and extract reaction rate coefficients of the system. We find good collisional stability of the mixture in the unitary regime, opening new possibilities for studies and probes of strongly interacting quantum gases in contact with a bath species.
We report the observation of a dimer-dimer inelastic collision resonance for ultracold Feshbach molecules made of bosonic sodium and rubidium atoms. This resonance, which we attribute to the crossing of the dimer-dimer threshold with a heteronuclear tetramer state, manifests itself as a pronounced inelastic loss peak of dimers when the interspecies scattering length between the constituent atoms is tuned. Near this resonance, a strong modification of the temperature dependence of the dimer-dimer scattering is observed. Our result provides insight into the heteronuclear four-body system consist…
We report on the creation of ultracold gases of bosonic Feshbach molecules of NaCs. The molecules are associated from overlapping gases of Na and Cs using a Feshbach resonance at 864.12(5) G. We characterize the Feshbach resonance using bound state spectroscopy, in conjunction with a coupled-channel calculation. By varying the temperature and atom numbers of the initial atomic mixtures, we demonstrate the association of NaCs gases over a wide dynamic range of molecule numbers and temperatures, reaching 70 nK for our coldest systems and a phase-space density near 0.1. This is an important stepp…
We investigate magnetoassociation of ultracold fermionic Feshbach molecules in a mixture of $^{40}$K and $^{87}$Rb atoms, where we can create as many as $7\times 10^4$ $^{40}$K$^{87}$Rb molecules with a conversion efficiency as high as 45%. In the perturbative regime, we find that the conversion efficiency depends linearly on the density overlap of the two gases, with a slope that matches a parameter-free model that uses only the atom masses and the known Feshbach resonance parameters. In the saturated regime, we find that the maximum number of Feshbach molecules depends on the atoms' phase-sp…
We show that radiofrequency (RF) radiation may be used to create Feshbach resonances in ultracold gases of alkali-metal atoms at desired magnetic fields that are convenient for atomic cooling and degeneracy. For the case of $^{39}$K+$^{133}$Cs, where there are no RF-free resonances in regions where Cs may be cooled to degeneracy, we show that a resonance may be created near 21~G with 69.2~MHz RF radiation. This resonance is almost lossless with circularly polarized RF, and the molecules created are long-lived even with plane-polarized RF.
Magnetic Feshbach resonances play a central role in experimental research of atomic gases at ultracold temperatures, as they allow one to control the microscopic interactions between ultracold atoms by tuning an applied magnetic field. These resonances arise due to strong hyperfine interactions between the unpaired electron and the nuclear magnetic moment of the alkali metal atoms. A major thrust of current research is to create an ultracold gas of diatomic alkali-metal molecules in the ground rovibrational state of the ground electronic $^1Σ$ state. Unlike alkali metal atoms, $^1Σ$ diatomic m…
Deactivation of broad quasi-one-dimensional dibosonic molecules is analyzed. Within integrable Lieb-Liniger-McGuire (LLMG) model an exact expression does not demonstrate suppression of the deactivation at low collision energies. Solution of Faddeev equations demonstrates that when a Feshbach resonance lifts the symmetry of the LLMG model the deactivation becomes suppressed. This effect shows a way for formation of a stable gas of dibosonic Feshbach molecules.
Two-color photoassociation of ground state $^6$Li$^{87}$Rb molecules via the $\mathrm{B}^1Π$ electronic state using short pulses near a magnetic Feshbach resonance is studied theoretically. A near-resonant magnetic field is applied to mix the hyperfine singlet and triplet components of the initial wave function and enhance the photoassociation rate, before the population is transferred to the ground state by a second pulse. We show that an increase of up to three orders of magnitude in the absolute number of molecules produced is attainable for deeply bound vibrational levels. This technique c…
We have performed radio-frequency dissociation spectroscopy of weakly bound ^6Li_2 Feshbach molecules using low-density samples of about 30 molecules in an optical dipole trap. Combined with a high magnetic field stability this allows us to resolve the discrete trap levels in the RF dissociation spectra. This novel technique allows the binding energy of Feshbach molecules to be determined with unprecedented precision. We use these measurements as an input for a fit to the ^6Li scattering potential using coupled-channel calculations. From this new potential, we determine the pole positions of t…
We present a detailed study of interspecies Feshbach resonances of the bosonic $^{23}\mathrm{Na}+$$^{39}\mathrm{K}$ mixture for magnetic fields up to $750 \, \mathrm{G}$ in various collision channels. A total of fourteen Feshbach resonances are reported, as well as four zero crossings of the scattering length and three inelastic two-body loss features. We use the observed magnetic field locations of the resonant features together with the known data on $^{23}\mathrm{Na}+$$^{40}\mathrm{K}$ to refine the singlet and triplet ground state potentials of NaK and achieve a consistent description of F…
We report on the observation of ultracold heteronuclear Feshbach molecules. Starting with a $^{87}$Rb BEC and a cold atomic gas of $^{85}$Rb, we utilize previously unobserved interspecies Feshbach resonances to create up to 25,000 molecules. Even though the $^{85}$Rb gas is non-degenerate we observe a large molecular conversion efficiency due to the presence of a quantum degenerate $^{87}$Rb gas; this represents a key feature of our system. We compare the molecule creation at two different Feshbach resonances with different magnetic-field widths. The two Feshbach resonances are located at $265…
We investigate phase separation of Bose-Einstein condensates (BECs) of two-component atoms and one-component molecules with a homonuclear Feshbach resonance. We develop a full model for dilute atomic and molecular gases including correlation of the Feshbach resonance and all kinds of interparticle interactions, and numerically calculate order parameters of the BECs in spherical harmonic oscillator traps at zero temperature with the Bogoliubov's classical field approximation. As a result, we find out that the Feshbach resonance can induce two types of phase separation. The actual phase structur…
We report the production of $^{87}$RbCs Feshbach molecules in a crossed-beam dipole trap. A mixture of $^{87}$Rb and $^{133}$Cs is cooled close to quantum degeneracy before an interspecies Feshbach resonance at 197\,G is used to associate up to $\sim5000$ molecules with a temperature of $\sim300$\,nK. The molecules are confined in the dipole trap with a lifetime of 0.21(1)\,s, long enough for future experiments exploring optical transfer to the absolute ground state. We have measured the magnetic moment of the Feshbach molecules in a magnetic bias field range between 181 and 185\,G to demonstr…
Ultracold molecules offer a broad variety of applications, ranging from metrology to quantum computing. However, forming "real" ultracold molecules, {\it i.e.} in deeply bound levels, is a very difficult proposition. Here, we show how photoassociation in the vicinity of a Feshbach resonance enhance molecular formation rates by several orders of magnitude. We illustrate this effect in heteronuclear systems, and find giant rate coefficients even in deeply bound levels. We also give a simple analytical expression for the photoassociation rates, and discuss future applications of the Feshbach Opti…
We study the rotational predissociation of atom - molecule complexes with very small binding energy. Such complexes can be produced by Feshbach resonance association of ultracold molecules with ultracold atoms. Numerical calculations of the predissociation lifetimes based on the computation of the energy dependence of the scattering matrix elements become inaccurate when the binding energy is smaller than the energy width of the predissociating state. We derive expressions that represent accurately the predissociation lifetimes in terms of the real and imaginary parts of the scattering length …
We model collisionless collective conversion of a degenerate Fermi gas into bosonic molecules via a Feshbach resonance, treating the bosonic molecules as a classical field and seeding the pairing amplitudes with random phases. A dynamical instability of the Fermi sea against association into molecules initiates the conversion. The model qualitatively reproduces several experimental observations {[Regal et al., Nature {\bf 424}, 47 (2003)]}. We predict that the initial temperature of the Fermi gas sets the limit for the efficiency of atom-molecule conversion.
Collisional resonances are an important tool which has been used to modify interactions in ultracold gases, for realizing novel Hamiltonians in quantum simulations, for creating molecules from atomic gases and for controlling chemical reactions. So far, such resonances have been observed for atom-atom collisions, atom-molecule collisions and collisions between Feshbach molecules which are very weakly bound. Whether such resonances exist for ultracold ground state molecules has been debated due to the possibly high density of states and/or rapid decay of the resonant complex. Here we report a v…
Magnetically tunable Feshbach resonances for polar paramagnetic ground-state diatomics are too narrow to allow for magnetoassociation starting from trapped, ultracold atoms. We show that non-resonant light can be used to engineer the Feshbach resonances in their position and width. For non-resonant field strengths of the order of $10^9\,$W/cm$^2$, we find the width to be increased by three orders of magnitude, reaching a few Gauss. This opens the way for producing ultracold molecules with sizeable electric and magnetic dipole moments and thus for many-body quantum simulations with such particl…
We have studied interspecies scattering in an ultracold mixture of $^{87}$Rb and $^{133}$Cs atoms, both in their lowest-energy spin states. The three-body loss signatures of 30 incoming s- and p-wave magnetic Feshbach resonances over the range 0 to 667 G have been catalogued. Magnetic field modulation spectroscopy was used to observe molecular states bound by up to 2.5 MHz$\times h$. Magnetic moment spectroscopy along the magneto-association pathway from 197 to 182 G gives results consistent with the observed and calculated dependence of the binding energy on magnetic field strength. We have c…
Magnetic Feshbach resonances are an invaluable tool for controlling ultracold atoms and molecules. They can be used to tune atomic interactions and have been used extensively to explore few- and many-body phenomena. They can also be used for magnetoassociation, in which pairs of atoms are converted into molecules by ramping an applied magnetic field across a resonance. Pairs of open-shell atoms, such as the alkalis, chromium, and some lanthanides, exhibit broad resonances because the corresponding molecule has multiple electronic states. However, molecules formed between alkali and closed-shel…
We observe thermalization in the production of a degenerate Fermi gas of polar ${}^{40}\text{K}{}^{87}\text{Rb}$ molecules. By measuring the atom--dimer elastic scattering cross section near the Feshbach resonance, we show that Feshbach molecules rapidly reach thermal equilibrium with both parent atomic species. Equilibrium is essentially maintained through coherent transfer to the ground state. Sub-Poissonian density fluctuations in Feshbach and ground-state molecules are measured, giving an independent characterization of degeneracy and directly probing the molecular Fermi--Dirac distributio…
We show that the photoassociation (PA) rate of ultracold diatomic molecules in the vicinity of a Feshbach resonance could be used to probe variations of the electron-to-proton mass ratio $β= m_e/m_p$, a quantity related to other fundamental constants via the $Λ_{\mathrm{QCD}}$ scale. The PA rate exhibits two features near a Feshbach resonance, a minimum and a maximum, which are very sensitive to the details of the interactions and the exact mass of the system. The effect and detection threshold are illustrated in the formation rates of ultracold Li$_2$ and LiNa molecules by performing coupled-…
The dynamical atom/molecule projection, recently used to probe fermion pairing, is fast compared to collective fermion times, but slow on the Feshbach resonance width scale. Theory of detuning-induced dynamics of molecules coupled to resonantly associating atom pairs, employing a time-dependent many-body Green's function approach, is presented. An exact solution is found, predicting a 1/3 power law for molecule production efficiency at fast sweep. The results for $s$- and p-wave resonances are obtained and compared. The predicted production efficiency agrees with experimental observations for …
Measurements of energy-resolved positron-molecule annihilation show the existence of positron binding and vibrational Feshbach resonances. The existing theory describes this phenomenon successfully for the case of infrared-active vibrational modes which allow dipole coupling between the incident positron and the vibrational motion. Presented here are measurements of positron-molecule annihilation made using a recently developed cryogenic positron beam capable of significantly improved energy resolution. The results provide evidence of resonances associated with infrared-inactive vibrational mo…
We demonstrate detection of NaRb Feshbach molecules at high magnetic field by combining molecular photodissociation and absorption imaging of the photofragments. The photodissociation process is carried out via a spectroscopically selected hyperfine Zeeman level correlated with the Na ($3P_{3/2}$) + Rb ($5S_{1/2}$) asymptote which, following spontaneous emission and optical pumping, leads to ground-state atoms in a single level with near unity probability. Subsequent to the dissociation, the number of molecules is obtained by detecting the resultant $^{23}$Na and $^{87}$Rb atoms. We have also …
We investigate magnetoassociation of ultracold Feshbach molecules from a Bose- Einstein condensate of Cs atoms and explore the spectrum of weakly bound molecular states close to the atomic threshold. By exploiting the variation of magnetic field experienced by a molecular cloud falling in the presence of a magnetic field gradient, we demonstrate the repeated output coupling of molecules from a single atomic cloud using a Feshbach resonance at 19.89 G. Using this method we are able to produce up to 24 separate pulses of molecules from a single atomic condensate, with a molecular pulse created e…
We study the association and dissociation dynamics of weakly bound heteronuclear Feshbach molecules using transverse RF-fields for expected parameters accessible through the microgravity environment of NASA's Cold Atom Laboratory (CAL) aboard the International Space Station, including temperatures at or below nK and atomic densities as low as $10^8$/cm$^3$. We show that under such conditions, thermal and loss effects can be greatly suppressed resulting in high efficiency in both association and dissociation of Feshbach molecules with mean size exceeding 10$^4a_0$, and allowing for the coherenc…
We study enhanced magneto-association of atoms into weakly-bound molecules near a Feshbach resonance using a quench preparatory stage. In anticipation of experiments with NASA's Cold Atom Laboratory aboard the International Space Station, we assume as a baseline a dual-species ($^{87}$Rb and $^{41}$K) gas in a parameter regime enabled by a microgravity environment. This includes subnanokelvin temperatures and dual-species gases at densities as low as 10$^8$/cm$^3$. Our studies indicate that, in such a regime, traditional magneto-association schemes are inefficient due to the weak coupling betw…
Enhanced positron annihilation on polyatomic molecules is a long-standing and complex problem. We report the results of calculations of resonant positron annihilation on methyl halides. A free parameter of our theory is the positron binding energy. A comparison with energy-resolved annihilation rates measured for CH$_3$F, CH$_3$Cl, CH$_3$Br [Barnes et al. Phys. Rev. A {\bf 74}, 012706 (2006)] shows good agreement and yields estimates of the binding energies.
Degenerate Fermi gas interacting with molecules near Feshbach resonance is unstable with respect to formation of a mixed state in which atoms and molecules coexist as a coherent superposition. Theory of this state is developed using a mapping to the Dicke model, treating molecular field in the single mode approximation. The results are accurate in the strong coupling regime relevant for current experimental efforts. The exact solution of the Dicke model is exploited to study stability, phase diagram, and nonadiabatic dynamics of molecular field in the mixed state.
In the vicinity of a Feshbach resonance, a system of ultracold atoms on an optical lattice undergoes rich physical transformations which involve molecule formation and hopping of molecules on the lattice and thus goes beyond a single-band Hubbard model description. We propose to probe the behavior of this system with a harmonic modulation of the magnetic field, and thus of the scattering length, across the Feshbach resonance, as an alternative to lattice-depth modulation spectroscopy. In the regime in which the single-band Hubbard model is still valid, we provide simulation data for this type …
We model the formation of stable heteronuclear molecules via pulsed Raman photoassociation of a two-component Bose-Einstein condensate near a strong Feshbach resonance, for both counterintuitive and intuitive pulse sequencing. Compared to lasers alone, weak Raman photoassociation is enhanced by as much as a factor of ten (five) for a counterintuitive (intuitive) pulse sequence, whereas strong Raman photoassociation is barely enhanced at all--regardless of pulse sequence. Stronger intra-atom, molecule, or atom-molecule collisions lead to an expected decrease in conversion efficiency, but strong…
We consider the possibilities for producing ultracold mixtures of K and Cs and forming KCs molecules by magnetoassociation. We carry out coupled-channel calculations of the interspecies scattering length for $^{39}$KCs, $^{41}$KCs and $^{40}$KCs and characterize Feshbach resonances due to s-wave and d-wave bound states, with widths ranging from below 1 nG to 5 G. We also calculate the corresponding bound-state energies as a function of magnetic field. We give a general discussion of the combinations of intraspecies and interspecies scattering lengths needed to form low-temperature atomic mixtu…
We consider the long-range states expected for complexes formed from an alkali-metal diatomic molecule in a singlet state and an alkali-metal atom. We explore the structure of the Hamiltonian for such systems, and the couplings between the six angular momenta that are present. We consider the patterns and densities of the long-range states, and the terms in the Hamiltonian that can cause Feshbach resonances when the states cross threshold as a function of magnetic field. We present a case study of $^{40}$K$^{87}$Rb+$^{87}$Rb. We show multiple types of resonance due to long-range states with ro…
In collisions at ultralow temperatures, molecules will possess Feshbach resonances, foreign to ultracold atoms, whose virtual excited states consist of rotations of the molecules. We estimate the mean spacing and mean widths of these resonant states, exploiting the fact the molecular collisions at low energy display chaotic motion. As examples, we consider the experimentally relevant molecules O_2, OH, and PbO. The density of s-wave resonant states for these species is quite high, implying that a large number of narrow resonant states will exist.
Feshbach resonances in ultra-cold atomic gases have led to some of the most important advances in atomic physics. They did not only enable ground breaking work in the BEC-BCS crossover regime [1], but are also widely used for the association of ultra-cold molecules [2], leading to the formation of molecular Bose-Einstein condensates [3,4] and ultra-cold dipolar molecular systems [5]. Here, we demonstrate the experimental realization of an optical Feshbach resonance using ultra-long range Rydberg molecules [6]. We show their practical use by tuning the revival time of a quantum many-body system…
Magnetically tunable Feshbach resonances were employed to associate cold diatomic molecules in a series of experiments involving both atomic Bose as well as two spin component Fermi gases. This review illustrates theoretical concepts of both the particular nature of the highly excited Feshbach molecules produced and the techniques for their association from unbound atom pairs. Coupled channels theory provides the rigorous formulation of the microscopic physics of Feshbach resonances in cold gases. Concepts of dressed versus bare energy states, universal properties of Feshbach molecules, as wel…
Scattering length, which can be measured in Bose-Einstein condensate and Feshbach molecule experiments, is extremely sensitive to the variation of fundamental constants, in particular, the electron-to-proton mass ratio (m_e/m_p or m_e/Lambda_{QCD}, where Lambda_{QCD} is the QCD scale). Based on single- and two-channel scattering model, we show how the variation of the mass ratio propagates to the scattering length. Our results suggest that variation of m_e/m_p on the level of 10^{-11}~10^{-14} can be detected near a narrow magnetic or an optical Feshbach resonance by monitoring the scattering …
Bound states and collisions of atoms with two-channel two-body interactions in harmonic waveguides are analyzed. The closed-channel contributions to two-atom bound states become dominant in the case of a weak resonance. At low energies and values of the non-resonant scattering length the problem can be approximated by a one-dimensional resonant model. Three-body problem becomes nonintegrable and the properties of triatomic molecules become different from those predicted by the integrable Lieb-Liniger-McGuire model.
The B-spline basis set method is applied to determining the rovibrational eigen-spectrum of diatomic molecules. A particular attention is paid to a challenging numerical task of an accurate and efficient description of the vibrational levels near the dissociation limit (halo-state and Feshbach molecules). Advantages of using B-splines are highlighted by comparing the performance of the method with that of the commonly-used discrete variable representation (DVR) approach. Several model cases, including the Morse potential and realistic potentials with 1/R^3 and 1/R^6 long-range dependence of th…
