The molecular structure and the analytical potential energy function of S2^- and S3^-

In this paper, the equilibrium geometry, harmonic frequency and dissociation energy of S2^- and S3^- have been calculated at QCISD/6-311＋＋G（3d2f） and B3P86/6-311＋＋G（3d2f） level. The S2^- ground state is of 2IIg, the S3^- ground state is of 2B1 and S3^- has a bent （C2v） structure with an angle of 115.65° The results are in good agreement with these reported in other literature. For S3^- ion, the vibration frequencies and the force constants have also been calculated. Base on the general principles of microscopic reversibility, the dissociation limits has been deduced. The Murrell-Sorbie potential energy function for S2^- has been derived according to the ab initio data through the least- squares fitting. The force constants and spectroscopic data for S2^- have been calculated, then compared with other theoretical data. The analytical potential energy function of S3^- have been obtained based on the many-body expansion theory. The structure and energy can correctly reappear on the potential surface.     

Hawking radiation of stationary and non-stationary Kerr–de Sitter black holes

Hawking radiation of the stationary Kerr–de Sitter black hole is investigated using the relativistic Hamilton–Jacobi method. Meanwhile, extending this work to a non-stationary black hole using Dirac equations and generalized tortoise coordinate transformation, we derived the locations, the temperature of the thermal radiation as well as the maximum energy of the non-thermal radiation. It is found that the surface gravity and the Hawking temperature depend on both time and different angles. An extra coupling effect is obtained in the thermal radiation spectrum of Dirac particles which is absent from thermal radiation of scalar particles. Further, the chemical potential derived from the thermal radiation spectrum of scalar particle has been found to be equal to the highest energy of the negative energy state of the scalar particle in the non-thermal radiation for the Kerr–de Sitter black hole. It is also shown that for stationary black hole space time, these two different methods give the same Hawking radiation temperature.     

The characteristics of doubly and triply charged ions C2^2＋, C2^2- and C2^3＋

Based on group theory and atomic and molecular reaction statics, this paper derives the possible electronic states of C^{2+}_2, C^{2-}_2 and C^{3+}_2, and their reasonable dissociation limits and determines their ground electronic states C^{2+}_2(X^3Π_u), C^{2-}_2(X^1Σ^+_g) and C^{3+}_2(X^4Σ^-_u) using quantum mechanical calculations at the level of QCISD/6-311G^*. All the potential energy curves of their ground states have both a minimum and a maximum, which are the so-called "energy trapped" molecules. This sort of potential maximum is chiefly due to Coulomb repulsion. We propose the perturbation effect of ionic charges, which is used to explain why the orbital degeneracy of diatomic ions may be removed. The characteristics of potential curves for diatomic ions are briefly described.     

The molecular structure and analytical potential energy function of HCO （X^2A＇）

In this paper the equilibrium structure of HCO has been optimized by using density functional theory （DFT）/ B3P86 method and CC-PVTZ basis. It has a bent （Cs, X^2A＇） ground state structure with an angle of 124.4095 °. The vibronic frequencies and force constants have also been calculated. Based on the principles of atomic and molecular reaction statics, the possible electronic states and reasonable dissociation limits for the ground state of HCO molecule have been determined. The analytic potential energy function of HCO （X^2A＇） molecule has been derived by using the many-body expansion theory. The contour lines are constructed, which show the static properties of HCO （X^2A＇）, such as the equilibrium structure, the lowest energies, etc. The potential energy surface of HCO （X^2A＇） is reasonable and very satisfactory.     

The structure and the analytical potential energy function of NH2 （X^2B1）

This paper reports that the equilibrium structure of NH2 has been optimized at the QCISD/6-311＋＋G （3df, 3pd） level. The ground-state NH2 has a bent （C2v, X^2B1） structure with an angle of 103.0582°. The geometrical structure is in good agreement with the other calculational and experimental results. The harmonic frequencies and the force constants have also been calculated. Based on the group theory and the principle of microscopic reversibility, the dissociation limits of NH2（C2v, X^2B1） have been derived. The potential energy surface of NH2（X^2B1） is reasonable. The contour lines are constructed, the structure and energy of NH2 reappear on the potential energy surface.     

Spin polarization effect for Tc2 molecule

Density functional method (DFT) (B3p86) of Gaussian98 has been used to optimize the structure of the Tc_2 molecule. The result shows that the ground state for Tc_2 molecule is an 11-multiple state and its electronic configuration is {}^{11}Σ_g^-, which shows the spin polarization effect of Tc_2 molecule of a transition metal element for the first time. Meanwhile, we have not found any spin pollution because the wavefunction of the ground state does not mingle with wavefunctions of higher energy states. So, that the ground state for Tc_2 molecule is an 11-multiple state is indicative of the spin polarization effect of Tc_2 molecule of a transition metal element: that is, there exist 10 parallel spin electrons. The non-conjugated electron is greatest in number. These electrons occupy different spacious tracks, so that the energy of Tc_2 molecule is minimized. It can be concluded that the effect of parallel spin of the Tc_2 molecule is larger than the effect of the conjugated molecule, which is obviously related to the effect of electron d delocalization. In addition, the Murrell--Sorbie potential functions with the parameters for the ground state {}^{11}Σ_g^- and other states of Tc_2 molecule are derived. Dissociation energy D_e for the ground state of T_{c2} molecule is 2.266eV, equilibrium bond length R_e is 0.2841nm, vibration frequency ω_e is 178.52cm^{-1}. Its force constants f_2, f_3, and f_4 are 0.9200aJ·nm^{-2}, --3.5700aJ·nm^{-3}, 11.2748aJ·nm^{-4} respectively. The other spectroscopic data for the ground state of Tc_2 molecule ω_eχ_e, B_e, α_e are 0.5523cm^{-1}, 0.0426cm^{-1}, 1.6331×10^{-4}cm^{-1} respectively.     

Monoenergetic electron parameters in a spheroid bubble model

A reliable analytical expression for the potential of plasma waves with phase velocities near the speed of light is derived.The presented spheroid cavity model is more consistent than the previous spherical and ellipsoidal models and it explains the mono-energetic electron trajectory more accurately,especially at the relativistic region.The maximum energy of electrons is calculated and it is shown that the maximum energy of the spheroid model is less than that of the spherical model.The electron energy spectrum is also calculated and it is found that the energy distribution ratio of electrons △E/E for the spheroid model under the conditions reported here is half that of the spherical model and it is in good agreement with the experimental value in the same conditions.As a result,the quasi-mono-energetic electron output beam interacting with the laser plasma can be more appropriately described with this model.     

Electric ignition energy evaluation and the energy distribution structure of energy released in electrostatic discharge process

Ignition energy is one of the important parameters of flammable materials, and evaluating ignition energy precisely is essential to the safety of process industry and combustion science and technology. By using electric spark discharge test system, a series of electric spark discharge experiments were conducted with the capacitor-stored energy in the range of 10 J, 100 J, and 1000 J, respectively. The evaluation method for energy consumed by electric spark, wire, and switch during capacitor discharge process has been studied respectively. The resistance of wire, switch, and plasma between electrodes has been evaluated by different methods and an optimized evaluation method has been obtained. The electric energy consumed by wire, electric switch, and electric spark-induced plasma between electrodes were obtained and the energy structure of capacitor-released energy was analyzed. The dynamic process and the characteristic parameters(the maximum power, duration of discharge process) of electric spark discharge process have been analyzed. Experimental results showed that, electric spark-consumed energy only accounts for 8%–14% of the capacitor-released energy. With the increase of capacitor-released energy, the duration of discharge process becomes longer, and the energy of plasma accounts for more in the capacitor-released energy. The power of electric spark varies with time as a damped sinusoids function and the period and the maximum value increase with the capacitor-released energy.     

Shell Correction at the Saddle Point for Superheavy Nucleus

The potential energy surface for superheavy nucleus has been studied within the framework of the constrained relativistic mean field theory, and the shell correction energy as a function of de[ormation has been extracted by the Strutinsky shell correction procedure. Contrary to the usual expectation, the shell correction energy at the saddle point is too important to be neglected, and it has essential contribution to the fission barrier in superheavy nucleus.     

The structure and the potential energy function of AlSO(C_S,X~2A″)

By using the B3P86/aug-cc-pvtz method,the accurate equilibrium geometry of the AlSO(CS,X2A″) molecule has been calculated and compared with available theoretical values.The obtained results show that the AlSO molecule has a most stable structure with bond lengths of R OAl = 0.1864 nm,R OS = 0.1623 nm,R AlS = 0.2450 nm,together with a dissociation energy of 13.88 eV.The possible electronic states and their reasonable dissociation limits for the ground state of the AlSO molecule were determined based on the principle of atomic and molecular reaction statics.The analytic potential energy function of the AlSO molecule was derived by the many-body expansion theory and the contour lines were constructed for the first time,which show the internal information of the AlSO molecule,including the equilibrium structure and stable point.The analysis demonstrates that the obtained potential energy function of AlSO is reasonable and successful and the present investigations provide important insights for further study on molecular reaction dynamics.     

Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

A new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form $ V(r,\theta ) = \tfrac{1} {8}\kappa r_e^2 \left( {\tfrac{r} {{r_e }} - \tfrac{{r_e }} {r}} \right)^2 + \tfrac{{\beta cos^2 \theta }} {{r^2 sin^2 \theta }} A new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form . The energy eigenvalues and eigenfunctions of the bound-states for the Schr?dinger equation in D-dimensions for this potential are obtained analytically by using the Nikiforov-Uvarov method. The radial and angular parts of the wave functions are obtained in terms of orthogonal Laguerre and Jacobi polynomials. We also find that the energy of the particle and the wave functions reduce to the energy and the wave functions of the bound-states in three dimensions.      

Weber potential from finite velocity of action?

The Weber potential energy U for charges q and q' separated by the distance R is U = (qq'/R)[1 – (dR/dt)²/2c²]. If this potential arises from a finite velocity c of energy transfer Q', where the retarded rate of transfer from q' to q is dQ(t-R/c)/dt = Q'[1 – (dR/dt)/c] and where the advanced rate from q to q' is dQ(t+R/c)/dt = Q'[1 + (dR/dt)/c], then the resultant time-average root-mean-square action is given by . Identifying Q' with the Coulomb potential energy qq'/R, the Weber potential is obtained. Using the same argument, Newtonian gravitation yields a corresponding Weber potential energy, qq'/R being replaced by ( - Gmm'/R).     

Spin polarization effect for Cr2 molecule

Density functional theory （DFT） （B3P86） of Gaussian 03 has been used to optimize the structure of the Cr2 molecule, a transition metal element molecule. The result shows that the ground state for the Cr2 molecule is a 13- multiple state, indicating that there exists a spin polarization effect in the Cr2 molecule. Meanwhile, we have not found any spin pollution because the wave function of the ground state does not mingle with wave functions of higher-energy states. So the ground state for Cr2 molecule being a 13-multiple state is indicative of spin polarization effect of the Cr2 molecule among transition metal elements, that is, there are 12 parallel spin electrons in the Cr2 molecule. The number of non-conjugated electrons is greatest. These electrons occupy different spatial orbitals so that the energy of the Cr2 molecule is minimized. It can be concluded that the effect of parallel spin in the Cr2 molecule is larger than the effect of the conjugated molecule, which is obviously related to the effect of electron d delocalization. In addition, the Murrell Sorbie potential functions with the parameters for the ground state and other states of the Cr2 molecule are derived. The dissociation energy De for the ground state of the Cr2 molecule is 0.1034eV, equilibrium bond length Re is 0.3396 nm, and vibration frequency we is 73.81cm^-1. Its force constants f2, f3 and f4 are 0.0835, -0.2831 and 0.3535 aJ. nm^-4 respectively. The other spectroscopic data for the ground state of the Cr2 molecule ωeχe, Be and αe are 1.2105, 0.0562 and 7.2938 x 10^-4cm^-1 respectively.     

Solution to the Schrödinger Equation for the Time-Dependent Potential

In this work, the Schrödinger equation with the time-dependent potential \(V(z,\hat{p},t)=g_{1}(t)z+g_{2}(t)\hat{p}+g_{3}(t)\) has been solved by the method of time-space transformation in 1+1 dimensions. The corresponding analytical wave function to Schrödinger equation is obtained. In addition, the discussion of solutions to particular cases has been made.     

N2O异构体结构与解析势能函数

在B3P86/cc-PVTZ水平上,对N₂O异构体进行优化计算,得出N₂O基态的单重态能量最低,其稳定构型为C_∞v构型,平衡核间距R₁=0.1121nm,R₂=0.1177nm,α＝180°,能量为-185.1188a.u.同时计算出基态的简正振动频率ω₁(Π)=601.5010 cm^{关键词： 异构体 多体项展式理论 解析势能函数}     

Potential energy curves and analytical potential energy functions of the metastable states of B2^＋＋

The multi-reference configuration interaction method and aug-cc-pvqz （AVQZ） have been used to calculate potential energy curves （PECs） of the singlet and triplet states of the riu and rig symmetry of B2＋＋. All of the four states （^l∏u, ^1∏g, ^3∏u and ^3∏g） are found to be metastable states, though the potential well of ^3∏u symmetry is very shallow. Based on the PECs, the analytical potential energy functions （APEFs） of these states have been fitted using the least square fitting method and two models of function. The spectroscopic parameters of each state are also calculated, and are compared with other investigations in the literature. The credibility and veracity of the two functions are evaluated. Some ideas to improve the fitting accuracy are presented. Also the vibrational levels for each state are predicted by solving the SchrSdinger equation of nuclear motion.     

PuNO分子结构与势能函数

本文运用密度泛函理论的B3LYP方法, 对钚原子应用LANL2DZ收缩价基函数, 氮、氧原子采用AUG-cc-pVTZ基函数, 分别对PuN, PuO, NO和PuNO体系进行了结构优化, 得到PuNO分子最稳构型为C_∞v (Pu-N-O), 电子态为⁶Σ^- (基态), 平衡核间距R_ON=0.12257 nm, R_NPu=0.22951 nm, 离解能D_e=8.10537 eV. 同时优化得到PuNO 分子存在两种亚稳态平衡构型分别为C_∞v (Pu-O-N), ⁶Σ^- (电子态)和C_s (O-Pu-N), A" (电子态), 以及分子体系相应的力学常数等. 拟合出PuN, PuO 和NO 分子的Murrell-Sorbie势能函数, 并使用多体项展式理论得到了PuNO分子的分析势能函数, 其等值势能图准确再现了PuNO分子最稳态构型及两个亚稳态构型的离解能和结构特性, 由此讨论了该分子体系的势能面静态特征.     

The splitting of low-lying states for hydroxyl molecule under spin--orbit coupling

The splitting of potential energy curves for the states $X^{2}\Pi _{3/2}$, $^{2}\Pi _{1/2}$ and $A^{2}\Sigma ^{ +}$ of hydroxyl OH under spin--orbit coupling (SOC) has been calculated by using the SO multi-configuration quasi-degenerate perturbation theory (SO-MCQDPT). Their Murrell--Sorbie (M--S) potential functions have been derived, then, the spectroscopic constants for $X^{2}\Pi _{3/2}$,$^{ 2}\Pi _{1/2}$ and $A^{2}\Sigma ^{ + }$ have been derived from the M--S function. The calculated dissociation energies for the three states are $D_{0}$[OH($X^{2}\Pi _{3/2})$]=34966.632cm$^{-1}$, $D_{0}$[OH($^{2}\Pi _{1/2})$]=34922.802cm$^{-1}$, and $D_{0}$[OH($A^{2}\Sigma ^{ + })$]=17469.794cm$^{-1}$, respectively. The vertical excitation energy $\nu [ {{ }^2\Pi _{1/2} ( {\nu = 0} ) \to {X}{ }^2\Pi _{3/2} ( {\nu = 0} )} ] = 139.6{\rm cm}^{-{\rm 1}}$. All the spectroscopic data for the $X^{2}\Pi _{3/2}$ and $^{2}\Pi _{1/2 }$ are given for the first time except the dissociation energy of $X^{2}\Pi _{3/2}$.