Na2 Molecule Research Articles

Potential parameters and eigen spectra of improved Scarf II potential energy function for diatomic molecules

In this paper, Scarf II potential is used to construct the improved Scarf II potential energy function (ISPEF) for applications to diatomic molecules. Conditions to be satisfied by a diatomic molecule potential are applied in the derivation of potential parameters of the ISPEF. Equation of bound state ro-vibrational energies is obtained for the ISPEF via a Pekeris-type approximation to the centrifugal term of the Schrödinger equation. The ISPEF and expression for eigen energies are used to model spectroscopic data of five diatomic molecules: , Na2 (c1 Πu), ScI (B1 Π), NaK (c3 Σ+) and CO (X1 Σ+). Average absolute deviation from dissociation energy (AAD) and mean absolute percentage deviation from Rydberg-Klein-Rees data (MAPD) are used to analyse data. On the basis of AAD as goodness-of-fit index, the ISPEF represents a near-perfect model for the diatomic molecules analysed and also a better model than the existing literature data of improved Tietz potential for the Na2 (c1 Πu) molecule. On the other hand, with the MAPD as goodness-of-fit index, the available literature results show that eigen energies of ISPEF are equivalent to ro-vibrational energies of Morse and improved generalised Pöschl-Teller potentials in fitting spectroscopic data of CO (X1 Σ+) molecule.

Solutions of the Schrödinger equation with improved Rosen Morse potential for nitrogen molecule and sodium dimer

The one-dimensional Schrödinger equation with an improved Rosen Morse potential energy function was studied in view of supersymmetric approach. The ro-vibrational energy equation and the corresponding unnormalized radial wave function were obtained. The variation of the vibrational energy with the screening parameter and the exponential term were studied graphically. Some vibrational expectation values were also calculated. Using the calculated vibrational energy equation, numerical values were generated for X1∑g+ state of N2 and 51Δg state of Na2 molecules respectively. The comparison of the results for the two molecules with the RKR data showed a very good agreement.

Selective excitation of vibrational states in three-state Na2 molecule by a chirped laser pulse

We have theoretically investigated the population transfer and the selective excitation of vibrational states of the target state in a three-state Na2 molecule using the chirped adiabatic passage. The population transfer among electronic states is studied. Compared with the single negative chirped pulse, complete population transfer can be achieved by using a positive chirped pulse. The effects of laser parameters on vibrational population distribution are discussed, and the selective excitation of vibrational states can be achieved by adjusting the central frequency. Finally, we give the method of selective excitation of specific vibrational states.

Three-player polaritons: nonadiabatic fingerprints in an entangled atom–molecule–photon system

A quantum system composed of a molecule and an atomic ensemble, confined in a microscopic cavity, is investigated theoretically. The indirect coupling between atoms and the molecule, realized by their interaction with the cavity radiation mode, leads to a coherent mixing of atomic and molecular states, and at strong enough cavity field strengths hybrid atom–molecule–photon polaritons are formed. It is shown for the Na2 molecule that by changing the cavity wavelength and the atomic transition frequency, the potential energy landscape of the polaritonic states and the corresponding spectrum could be changed significantly. Moreover, an unforeseen intensity borrowing effect, which can be seen as a strong nonadiabatic fingerprint, is identified in the atomic transition peak, originating from the contamination of the atomic excited state with excited molecular rovibronic states.

Relativistic spectral bounds for the general molecular potential: application to a diatomic molecule.

We tackle with the Dirac equation in the presence of the general molecular potential (GMP). The spin symmetric solution of the relativistic wave equation is obtained by considering the Pekeris-type approximation scheme to deal with the centrifugal term, and in order to solve second-order differential equation, the asymptotic iteration method (AIM) is used. The closed form of the energy eigenvalue equation is found out for any values of the angular momentum quantum number. We calculate the relativistic vibrational bound state energies of 51Δg state of Na2 molecule and compare them with the Rydberg-Klein-Rees (RKR) data. We show that relativistic vibrational energies of this molecule, which are found in the spin symmetric case, are more convenient with experimental RKR data than non-relativistic vibrational energies. We also obtain normalization constant by considering inductive approach and investigate the radial eigenfunctions and probability density functions corresponding to different eigenvalues graphically for the Na2(51Δg) molecule.

Time-resolved double-resonance spectroscopy: Lifetime measurement of the electronic state of molecular sodium.

We report on the lifetime measurement of the state of Na2 molecules, produced in a heat-pipe oven, using a time-resolved spectroscopic technique. The level was populated by two-step two-color double resonance excitation via the intermediate state. The excitation scheme was done using two synchronized pulsed dye lasers pumped by a Nd:YAG laser operating at the second harmonics. The fluorescence emitted upon decay to the final state was measured using a time-correlated photon counting technique, as a function of argon pressure. From this, the radiative lifetime was extracted by extrapolating the plot to collision-free zero pressure. We also report the calculated radiative lifetimes of the ro-vibrational levels in the range of = 0-200 with J = 1 and J = 31 using the LEVEL program for bound-bound and the BCONT program for bound-free transitions. Our calculations reveal the importance of the bound-free transitions on the lifetime calculations and a large difference of about a factor of three between the J = 1 and J = 31 for the = 40 and = 100, respectively, due to the wavefunction alternating between having predominantly inner and outer well amplitude.

Conical Intersections Induced by Quantum Light: Field-Dressed Spectra from the Weak to the Ultrastrong Coupling Regimes.

In classical laser fields with frequencies resonant with the electronic excitation in molecules, it is by now known that conical intersections are induced by the field and are called light-induced conical intersections (LICIs). As optical cavities have become accessible, the question arises whether their quantized modes could also lead to the appearance of LICIs. A theoretical framework is formulated for the investigation of LICIs of diatomics in such quantum light. The eigenvalue spectrum of the dressed states in the cavity is studied, putting particular emphasis on the investigation of absorption spectra of the Na2 molecule, that is, on the transitions between dressed states, measured by employing a weak probe pulse. The dependence of the spectra on the light-matter coupling strength in the cavity and on the frequency of the cavity mode is studied in detail. The computations demonstrate strong nonadiabatic effects caused by the appearing LICI.

Relativistic spinless energies and thermodynamic properties of sodium dimer molecule

Relativistic spinless rotational-vibrational energy eigenvalue relation has been obtained by solving the Klein-Gordon equation in the presence of General Molecular potential (GMP). The relativistic energy eigenvalue relation has been reduced to the non-relativistic one in the non-relativistic limit. Relativistic and non-relativistic vibrational energies have been computed for the 51Δg state of Na2 molecule and compared with Rydberg-Klein-Rees (RKR) data. It has been concluded that depending on relation between scalar and vector potentials, relativistic effect can cause a slight decrease in the vibrational energies or can lead to an increase. In addition, thermodynamic properties such as vibrational mean energy U, vibrational specific heat C, vibrational free energy F and vibrational entropy S for Na2(51Δg) molecule have been examined with respect to temperature and the values of principal quantum number.

Development of polarization consistent basis sets for spin-spin coupling constant calculations for the atoms Li, Be, Na, and Mg.

The pcJ-n basis set, optimized for spin-spin coupling constant calculations using density functional theory methods, are expanded to also include the s-block elements Li, Be, Na, and Mg, by studying several small molecules containing these elements. This is done by decontracting the underlying pc-n basis sets, followed by augmentation with additional tight functions. As was the case for the p-block elements, the convergence of the results can be significantly improved by augmentation with tight s-functions. For the p-block elements, additional tight functions of higher angular momentum were also needed, but this is not the case for the s-block elements. A search for the optimum contraction scheme is carried out using the criterion that the contraction error should be lower than the inherent error of the uncontracted pcJ-n relative to the uncontracted pcJ-4 basis set. A large search over possible contraction schemes is done for the Li2 and Na2 molecules, and based on this search contracted pcJ-n basis sets for the four atoms are recommended. This work shows that it is more difficult to contract the pcJ-n basis sets, than the underlying pc-n basis sets. However, it also shows that the pcJ-n basis sets for Li and Be can be more strongly contracted than the pcJ-n basis sets for the p-block elements. For Na and Mg, the contractions are to the same degree as for the p-block elements.

Direct Signatures of Light-Induced Conical Intersections on the Field-Dressed Spectrum of Na2.

Rovibronic spectra of the field-dressed homonuclear diatomic Na2 molecule are investigated to identify direct signatures of the light-induced conical intersection (LICI) on the spectrum. The theoretical framework formulated allows the computation of the (1) field-dressed rovibronic states induced by a medium-intensity continuous-wave laser light and the (2) transition amplitudes between these field-dressed states with respect to an additional weak probe pulse. The field-dressed spectrum features absorption peaks resembling the field-free spectrum as well as stimulated emission peaks corresponding to transitions not visible in the field-free case. By investigating the dependence of the field-dressed spectra on the dressing-field wavelength, in both full- and reduced-dimensional simulations, direct signatures of the LICI can be identified. These signatures include (1) the appearance of new peaks and the splitting of peaks for both absorption and stimulated emission and (2) the manifestation of an intensity-borrowing effect in the field-dressed spectrum.

Steering population transfer of the Na2 molecule by an ultrashort pulse train

We theoretically investigate the complete population transfer among quantum states of the molecule using ultrashort pulse trains using the time-dependent wave packet method. The population accumulation of the target state can be steered by controlling the laser parameters, such as the variable pulse pairs, the different pulse widths, the time delays and the repetition period between two contiguous pulses; in particular, the pulse pairs and the pulse widths have a great effect on the population transfer. The calculations show that the ultrashort pulse train is a feasible solution, which can steer the population transfer from the initial state to the target state efficiently with lower peak intensities.

Probability and Flux Densities in the Center-of-Mass Frame.

For an arbitrary nonstationary wave function of a nonrelativistic closed many-body system consisting of arbitrary interacting particles, the general expressions for the time-dependent one-particle probability and flux densities in the center-of-mass frame without applying Born-Oppenheimer approximation are obtained. Even the wave function for the translation is additionally introduced; it disappears in the center-of-mass frame automatically. It is shown that for the rotational ground state the time-dependent probability and flux densities of an arbitrary particle in the center-of-mass frame are isotropic. It means that the angular dependence is absent but these densities depend on radius and time. More importantly, it is shown that the angular components of the time-dependent flux density vanish. With these statements, one can calculate the radial component of the radius- and time-dependent electronic flux density within the Born-Oppenheimer approximation via the continuity equation. Application of this theory to the pulsating or exploding "quantum bubble" of the vibrating or dissociating Na2 molecule in the rotational ground state, respectively, is found elsewhere in this issue.

Steering population transfer between electronic states of the Na2 molecule beyond the rotating wave approximation

By utilizing the time-dependent wavepacket method, we studied the population transfer between the electronic states of the Na2 molecule driven by femtosecond laser pulses. Three electronic states, , and , are taken to be the initial, the intermediate and the target states, respectively. Two classes of calculations are performed for comparison: one is to invoke the rotating wave approximation (RWA), and the other is more accurate calculations without consideration of RWA (non-RWA). It is found that for this three-state-model, the RWA fails to describe the accurate population transfer process in femtosecond-pulse fields and that the adiabatic passage is hard to be constructed in the non-RWA calculations. Based on the non-RWA calculations, we determined a set of laser parameters, including the peak intensity, detuning, pulse duration and time delay, for efficient population transfer.

Controlling vibrational population distribution of the Na2 molecule by ultrashort pulses in the three-state system

The population transfer and vibrational population distribution of the target state in the three-state Na2 molecule are theoretically investigated by using the time-dependent wave packet method. The population can be efficiently transferred to the target state by the process of adiabatic passage by light induced potentials (APLIP). The vibrational population distribution of the target state can be controlled by adjusting the laser parameters, such as the pulse width, time delay, intensity and envelope. The calculations show the vibrational population distribution variations under different laser parameters where the preparation of the specific state for molecules can be rationalised. In addition, the influence of the ionization state on the population transfer is studied.

Thermodynamic properties for the sodium dimer

We present a closed-form expression of the classical vibrational partition function for the improved Rosen-Morse potential energy model. We give explicit expressions for the vibrational mean energy, vibrational specific heat, vibrational free energy, and vibrational entropy for diatomic molecule systems. The properties of these thermodynamic functions for the Na2 dimer are discussed in detail. We find that the improved Rosen-Morse potential model is superior to the harmonic oscillator in calculating the heat capacity for the Na2 molecules.

Modelling of diatomic molecules

ABSTRACTA new general molecular potential is introduced to model diatomic molecular structures. Eigenfunctions and rotational-vibrational energy eigenvalue equation of the Schrödinger particle in the presence of centrifugal term are obtained for the general molecular potential. By using this rotational-vibrational energy eigenvalue equation, we calculate vibrational energy eigenvalues of 7Li2(6 1Πu), SiC(X 3Π), , ScI(X 1Σ+), and Na2(5 1Δg) molecules and the results are compared with Rydberg–Klein–Rees (RKR) data. We show that the general molecular potential is very convenient for diatomic molecules in fitting RKR data. Also, we show that the general molecular potential can be reduced to Rosen–Morse and Trigonometric Pöschl–Teller potentials which are used for some vibrations of some polyatomic molecules such as NH3 and SO2.

Molecular spinless energies of the improved Rosen-Morse potential energy model in D dimensions

Under the circumstance of equal scalar and vector potentials, we solve the Klein-Gordon equation with the improved Rosen-Morse potential energy model in D spatial dimensions. The relativistic bound state energy equation has been obtained by using the supersymmetric WKB approximation approach. For fixed vibrational quantum number and various rotational quantum numbers, the relativistic energies for the 33Σg+ state of the Cs2 molecule and the 51Δg state of the Na2 molecule increase as D increases. We observe that the behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Dynamical coupling of plasmons and molecular excitations by hybrid quantum/classical calculations: time-domain approach

The presence of plasmonic material influences the optical properties of nearby molecules in untrivial ways due to the dynamical plasmon-molecule coupling. We combine quantum and classical calculation schemes to study this phenomenon in a hybrid system that consists of a Na2 molecule located in the gap between two Au/Ag nanoparticles. The molecule is treated quantum-mechanically with time-dependent density-functional theory, and the nanoparticles with quasistatic classical electrodynamics. The nanoparticle dimer has a plasmon resonance in the visible part of the electromagnetic spectrum, and the Na2 molecule has an electron-hole excitation in the same energy range. Due to the dynamical interaction of the two subsystems the plasmon and the molecular excitations couple, creating a hybridized molecular-plasmon excited state. This state has unique properties that yield e.g. enhanced photoabsorption compared to the freestanding Na2 molecule. The computational approach used enables decoupling of the mutual plasmon-molecule interaction, and our analysis verifies that it is not legitimate to neglect the backcoupling effect when describing the dynamical interaction between plasmonic material and nearby molecules. Time-resolved analysis shows nearly instantaneous formation of the coupled state, and provides an intuitive picture of the underlying physics.

D-dimensional energies for sodium dimer

We solve the Schrödinger equation with the improved Tietz potential energy model in D spatial dimensions. The D-dimensional rotation–vibrational energy spectra have been obtained by using the supersymmetric shape invariance approach. The rotation–vibrational energies for the A1∑u+ and C1Пu states of the Na2 molecule increase as D increases in the presence of a fixed vibrational quantum number and rotational quantum number. It is observed that the behavior of the vibrational energies in higher dimensions remains similar to that of the three-dimensional system. We find that the D-dimensional scaling method resembles a translation transformation from the higher dimensions to the three dimensions.

High-pressure synthesis and structural characterization of the type II clathrate compound Na(30.5)Si(136) encapsulating two sodium atoms in the same silicon polyhedral cages.

Single crystals of sodium containing silicon clathrate compounds Na8Si46 (type I) and NaxSi136 (type II) were prepared from the mixtures of NaSi and Si under high-pressure and high-temperature conditions of 5 GPa at 600-1000 °C. The type II crystals were obtained at relatively low-temperature conditions of 700-800 °C, which were found to have a Na excess composition Na30.5Si136 in comparison with the compounds NaxSi136 (x ≤ 24) obtained by a thermal decomposition of NaSi under vacuum. The single crystal study revealed that the Na excess type II compound crystallizes in space group Fd3̅m with a lattice parameter of a = 14.796(1) Å, slightly larger than that of the ambient phase (Na24Si136), and the large silicon hexakaidecahedral cages (@Si28) are occupied by two sodium atoms disordered in the two 32e sites around the center of the @Si28 cages. At temperatures <90 K, the crystal symmetry of the compound changes from the face-centered to the primitive cell with space group P213, and the Na atoms in the @Si28 cages are aligned as Na2 pairs. The temperature dependence of the magnetic susceptibility of Na30.5Si136 suggests that the two Na ions (2 Na(+)) in the cage are changed to a Na2 molecule. The Na atoms of Na30.5Si136 can be deintercalated from the cages topochemically by evacuation at elevated temperatures. The single crystal study of the deintercalated phases NaxSi136 (x = 25.5 and 5.5) revealed that only excess Na atoms have disordered arrangements.