原子的解析波函数——Ⅱ.第二周期元素的正常态原子与离子的解析波函数与能量积分的计算

我们在文献[1]中设计了一套五个参数的变分波函数用来计算了周期表中前面十个原子的能量,所得结果比过去一些作者用四参数波函数所算得的结果为好。我们在过去计算经验的基础上,另外找到了一套特别简单的解析波函数,其形式为1s电子:ψ₁(r)=N₁e^-μar,2s电子:ψ₂(r)=N₂[(μr)e^-μr-Ne^-μar],2p电子:ψ₃(r)=N₃(μr)cosθe^-μr,ψ₄(r)=N₄(μr)sinθe^iφ-μr,ψ₅(r)=N₅(μr)sinθe^-iφ-μr,式中的α与μ为变分参数;N₁,N₂,N₃,N₄,N₅为归一化因子;N为正交化系数。μ可用解析法来决定,因而只有一个参数α要由数值法来决定。我们用这样的波函数算出了第二周期元素的正常态原子和离子(共有八十几个原子态)的各电子的各种能量积分值及总能量值,并确定了波函数的最佳参数值。其结果与五参数波函数的计算结果相比,一般相差在万分之一至千分之一的范围内,并比最近有些作者用一种三参数波函数所算的结果还好。根据这些结果,我们还讨论了Slater近似计算法的可靠程度和适用范围。     

原子的解析波函数

我们设计了一套变分波函数,用来计算了周期表中前面十个原子的能量。我们设计的单电子试探波函数具有下列形式:1s:ψ₁(r)=N₁e^-μαr[1+(μbr)²], 2s:ψ₂(r)=N₂[(μr)e^-μr-Ne^-μcr], 2p:ψ₃(r)=N₃(μdr)cosθe^-μdr, ψ₄(r)=N₄(μdr)sinθe^iφ-μdr, ψ₅(r)=N₅(μdr)sinθe^-iφ-μdr。式中的a,b,c,d及μ为五个变分参数。N₁,N₂,N₃,N₄与N₅为归一化因子;N由ψ₁与ψ₂的正交条件来决定。用这种波函数来计算原子的能量,所得的结果比莫尔斯等人(P.M.Morse,L.A.Young and E.S.Haurwitz)用他们设计的四参数波函数所算得的结果为好,更接近实验值,同时也接近于由自洽场所算出的结果。若我们的波函数中固定c等于1不变,这时就变为只有四个参数的波函数,结果仍比莫尔斯等人的好。     

四态叠加多模叠加态光场|Ψe(4),Ⅲ〉q的等阶N次方Y压缩

根据量子力学的线性叠加原理,构造了由多模偶相干态与多模虚偶相干态组成的第Ⅲ种四态叠加多模叠加态光场态|Ψ_e⁽⁴⁾,Ⅲ〉_q的等阶N次方Y压缩特性.结果发现:1) 当压缩阶数N=4m,(m=1,2,3,…)时,态|Ψ_e⁽⁴⁾,Ⅲ〉_q恒处于等阶数N-Y最小测不准态;2) 当压缩阶数N=4m′+2,(m′=0,1,2,…)时,在(θ₁-θ₂),q,R_j,r₁,r₂等取不同的组合定值下,态|Ψ_e⁽⁴⁾,Ⅲ〉_q可分别呈现出等阶N次方Y压缩效应与"半相干态"效应;3) 当压缩阶数N为奇数时,在(θ₁-θ₂),q,R_j,r₁,r₂等取不同的组合定值下,态|Ψ_e⁽⁴⁾,Ⅲ〉_q可呈现出等阶N次方Y压缩效应.     

Wei Hua’s four-parameter potential: Comments and computation of molecular constants αe and εeXe

The value of adjustable parameterC in the four-parameter potentialU(r) =D _e [(1 - exp[-b(r -r _e)])/(1 -C exp[-b(r -r _e)])]² has been expressed in terms of molecular parameters and its significance has been brought out. The potential so constructed, withC derived from the molecular parameters, has been applied to ten electronic states in addition to the states studied by Wei Hua. Average mean deviation for these 25 states has been found to be 3.47 as compared to 6.93, 6.95 and 9.72 obtained from Levine, Varshni and Morse potentials, respectively. Also Dunham’s method has been used to express rotation-vibration interaction constant (α_e) and anharmonicity constant (ω_ex_e) in terms ofC and other molecular constants. These relations have been employed to determine these quantities for 37 electronic states. For α_e, the average mean deviation is 7.2% compared to 19.7% for Lippincott’s potential which is known to be the best to predict these values. Average mean deviation for (ω_ex_e) turns out to be 17.4% which is almost the same as found from Lippincott’s potential function.     

Proton shielding function for hydrogen chloride

For a diatomic molecule the nuclear shielding constant σ(ξ) of either nucleus can be expanded as a power series in the relative displacement from equilibrium ξ. Thus the nuclear shielding function is

where ξ=(r-r _e)/r _e with r the actual bond length and r _e the equilibrium bond length. The σ_e ⁽ⁱ⁾ are molecular parameters. By experimental observation of the temperature dependence of the proton magnetic shielding of hydrogen chloride gas it is possible, after allowing for intermolecular effects, to obtain the values σ_e ⁽⁰⁾=32·48 (±0·33) p.p.m. and σ_e ⁽¹⁾=-100 (±24) p.p.m. for the coefficients of the proton shielding function. Using this data it is possible to show that the isotope shift between H³⁷Cl and H³⁵Cl is about 0·001 p.p.m. By a comparison with earlier results for molecular hydrogen it would appear that in some instances differences in vibrational and rotational averaging may alter chemical shifts between different compounds by amounts considerably larger than the experimental error in chemical shift measurement.     

The bond force constants of graphene and benzene calculated by density functional theory

Stretching (k_r) and bending (k_θ) bond force constants appropriate to describe the bond stiffness of graphene and benzene are calculated using density functional theory. The effect of employing different exchange-correlation functionals for the calculation of k_r and k_θ is discussed using the generalised gradient approximation (GGA) and the local density approximation (LDA). For benzene, k_r = 7.93 mdyn Å^-1 and k_θ = 0.859 mdyn Å rad^-2 using LDA, while k_r = 7.67 mdyn Å^-1 and k_θ = 0.875 mdyn Å rad^-2 using GGA. For graphene, k_r = 7.40 mdyn Å^-1 and k_θ = 0.769 mdyn Å rad^-2 using LDA, while k_r = 6.88 mdyn Å^-1 and k_θ = 0.776 mdyn Å rad^-2 using GGA. This means the difference between the bond force constants for benzene and graphene can be as large as ～12%. The comparison between these two systems allows for elucidation of the effect of periodicity and substitution of carbon atoms by hydrogen in the stiffness of C–C bonds. This effect can be explained by a different redistribution of the charge density when the systems are subjected to strain. The parameters k_r and k_θ computed here can serve as an input to molecular mechanics or finite element codes of larger carbon molecules, which in the past had frequently assumed the same bond force constants for graphene, benzene or carbon nanotubes.     

Theoretical study of stereodynamics for the O（3P） ＋ H2 → OH ＋ H reaction

This paper employs the quasi-classical trajectory calculations to study the influence of collision energy on the title reaction on the potential energy surface of the ground ³A' triplet state developed by Rogers et al. (J. Phys. Chem. A 2000 104 2308). It calculates the product angular distribution of P(θ_r), P(φ_r) and P(θ_r, φ_r) which reflects vector correlation. The distribution P(θ_r) shows that product rotational angular momentum vectors j' of the products are strongly aligned along the relative velocity direction k. The distribution of P(φ_r) implies a preference for left-handed product rotation in planes parallel to the scattering plane. Four different polarisation-dependent cross-sections are also presented in the centre-of-mass frame. Results indicate that OH is sensitively affected by collision energies of H₂.     

Toward a theory of the third-order elastic modulus of crystals with A2 structure: The case of α-Fe

The third-order elastic modulus of α-Fe were calculated based on the computation of lattice sums. The lattice sums were determined using an integer rational basis of invariants composed by vectors connecting equilibrium atomic positions in the crystal lattice. Irreducible interactions within clusters consisting of atomic pairs and triplets were taken into account in performing the calculations. Comparison with experimental data showed that the potential can be written in the form of e₉ = - ?_i,k A₁₉ r_ik ^{- 6} + ?_i,k A₂₉ r_ik ^{- 12} + ?_i,k,l Q₉ I₉ ^{- 1}\varepsilon _9 = - \sum\nolimits_{i,k} {A_{19} r_{ik}^{ - 6} } + \sum\nolimits_{i,k} {A_{29} r_{ik}^{ - 12} + \sum\nolimits_{i,k,l} {Q_9 I_9^{ - 1} } }, where I₉ = [(r)\vec]_ik² [ ( [(r)\vec]_ik [(r)\vec]_kl )² + ( [(r)\vec]_li [(r)\vec]_ik )² ] + [(r)\vec]_kl² [ ( [(r)\vec]_ik [(r)\vec]_kl )² + ( [(r)\vec]_kl [(r)\vec]_li )² ] + [(r)\vec]_li² [ ( [(r)\vec]_li [(r)\vec]_ik )² + ( [(r)\vec]_kl [(r)\vec]_li )² ]I_9 = \vec r_{ik}^2 \left[ {\left( {\vec r_{ik} \vec r_{kl} } \right)^2 + \left( {\vec r_{li} \vec r_{ik} } \right)^2 } \right] + \vec r_{kl}^2 \left[ {\left( {\vec r_{ik} \vec r_{kl} } \right)^2 + \left( {\vec r_{kl} \vec r_{li} } \right)^2 } \right] + \vec r_{li}^2 \left[ {\left( {\vec r_{li} \vec r_{ik} } \right)^2 + \left( {\vec r_{kl} \vec r_{li} } \right)^2 } \right]. If the values of [(r)\vec]_ik\vec r_{ik} are scaled in half-lattice constant units, then A₁₉ = 1.22 ë t⁹ û GPa, A₂₉ = 5.07 ×10² ë t¹⁵ û GPa, Q₉ = 5.31 ë t⁹ û GPaA_{19} = 1.22\left\lfloor {\tau ^9 } \right\rfloor GPa, A_{29} = 5.07 \times 10^2 \left\lfloor {\tau ^{15} } \right\rfloor GPa, Q_9 = 5.31\left\lfloor {\tau ^9 } \right\rfloor GPa, and τ = 1.26 ?. It is shown that the condition of thermodynamic stability of a crystal requires that we allow for irreducible interactions in atom triplets in at least four coordination spheres. The analytical expressions for the lattice sums determining the contributions from irreducible interactions in the atom triplets to the second- and third-order elastic moduli of cubic crystals in the case of interactions determined by I ₉ are presented in the appendix.     

Hypervirial perturbation theory for eigenenergies for two types of atomic potentials

Eigenenergies are calculated for the potentialsV ₁(r)=−(a/r)[1+(1+br)e^−2br ] andV ₂(r)=−(v/r)[1 −λr(1−Z ⁻¹)(1+λr)⁻¹], using renormalized series technique. Accurate results produced here for various eigenstates agree with those available in the literature.     

Interpreting the data on helium-ion scattering in metallic films

Earlier measurements of angular distributions in the multiple scattering of a helium-ion beam with energy below 300 keV on Al yielded rather unexpected results: the ratio between the half-width of the measured angular distribution, (θ_1/2)_e, and that predicted with the Moliere–Bethe theory, (θ_1/2)_MB, proved to stay almost constant throughout the investigated energy range. At the same time, one could expect the (θ_1/2)_e/(θ_1/2)_MB value to be affected by the beam-content variation due to the charge–exchange scattering. Towards resolving this problem, we compute the interaction potentials between the Не⁺⁺, Не⁺, and Не⁰ ions and the scattering atoms and reveal the electron-screening effects on the scattering process. Thereby, we explain the energy dependence of (θ_1/2)_e/(θ_1/2)_MB observed in the old and new measurements carried out at higher beam energies.     

碰撞能及反应物振动激发对Ar+H_2~+→ArH~++H反应立体动力学性质的影响

基于我们最近所构建的Ar+H+2→ArH++H(12A′)反应的新势能面,采用准经典轨线法研究了碰撞能分别为0.48,0.77,1.24 eV以及能量为0.48 eV时反应物不同振动态下Ar+H+2→ArH++H反应的立体动力学性质.结果显示在给定的碰撞能情况下,以及当反应物振动量子数由0变到2时计算的积分反应截面与实验值符合得较好.通过比较发现,碰撞能对此反应k-j′关联函数P(θr)分布的影响大于其受振动激发的影响,并且关于k-k′-j′三矢量相关的函数P(?r)分布以及极化微分反应截面对碰撞能较敏感,同时发现振动激发对P(?r)分布和极化微分反应截面也有较大的影响.     

Influence of reagent vibration on the stereodynamics of the Li + HF → LiF + H reaction

We investigate the influence of reagent vibration on the stereodynamics of the title reaction by the quasi-classical trajectory on the Aguado-Paniagua2-potential energy surface developed by Aguado et al. (J. Chem. Phys. 1997 106 1013). The cross sections and reaction probability as functions of the reagent vibration are calculated in the centre-of-mass frame. The product angular distributions of p(θ_r), p(φ_r), and p(θ_r, φ_r), which reflect the vector correlation, are also presented and discussed. The results indicate that the vector properties are sensitively affected by the vibrational excitation.     

Theoretical study of stereodynamics for the reaction O(3P) +D2 (v=0, j=0) to OD+D and isotope effect

Quasi-classical trajectory (QCT) calculations have been performed to study the product polarization behaviours in the reaction O(³P) + D₂ (v= 0, j= 0)→OD + D. By running trajectories on the ³A′ and ³A″ potential energy surfaces (PESs), vector correlations such as the distributions of the polarization-dependent differential cross sections (PDDCSs), the angular distributions of P(θ_r) and P(ø_r) are presented. Isotope effect is discussed in this work by a comprehensive comparison with the reaction O(³P) + H₂ (v= 0, j= 0) → H + H. Common characteristics as well as differences are discussed in product alignment and orientation for the two reactions. The isotope mass effect differs on the two potential energy surfaces: the isotope mass effect has stronger influence on P(θ_r) and PDDCSs of the ³A′ PES while the opposite on P(ø_r) of the ³A″ potential energy surface.     

Dissociation energy of diatomic molecules

The dissociation energy of twelve diatomic molecules has been determined by fitting four-parameter potential functionU(r)=D _e[[1−exp{−b(r−r _e)}]/ [1−Cexp{−b(r−r _e)}]]² to the true Rydberg-Klein-Rees (RKR) curves for their fifteen electronic states using the mean square deviation as the criterion for the selection of the best fit. Average deviation ofD _e has been found to be 2.7% as compared to 20.5% obtained with Lippincott’s potential function for these molecules. In addition the anharmonocity constantω _ex_e has also been calculated for the same electronic states yielding average mean deviation 8.9%.     

Utilization of rotational and vibrational temperatures for plasma diagnostics of high-pressure discharges

In the spectra of high-pressure discharges excited in molecular gases, very intensive molecular spectral bands may usually be observed. We may determine the rotational and vibrational temperatures without difficulty, however, the rotational and vibrational temperatures (T _r, T_v) do not offen equal to the temperature of neutral gas (T ₀) or to that of electrons (T _e). If the collision cross sections of electronic, atomic, and molecular excitation (deexcitation) are known, we may then calculate the dependence of the rotational and vibrational temperatures onT _e,T ₀,N _e and the pressure of the gas. The calculations have been performed for pure N₂ and for an Ar-N₂ mixture at atmospheric pressure. The computed graphs make it possible to determine some of the values 4T _e,T ₀,N _e if the temperaturesT _r andT _v are known.The author wishes to extend his thanks to Prof. V. Truneek for valuable comments and to Mr. A. Struka for the preparation of the diagrams.     

Comment on: “Universal relation between spectroscopic constants”

Kaur and Mahajan [1] have claimed to derive a universal relation InG = 1.91578(±0.09727) + 0.97111(±0.03809) In Δ between the Sutherland parameter Δ(=ω _er _e ² /2D_e) and the dimensionless parameterG(= 8ω _ex_e/B_e) for the ground as well as excited electronic states of diatomic molecules. Validity of this relation is checked and we find that the relation is not correct. Next, we checked the validity of the relation Δ = 2.2r_e for the alkali group diatomic molecules. This relation is also found not to be correct.     

Coupled cluster calculations for the interstellar molecule HC3NH+

An accurate equilibrium structure has been established for the linear interstellar molecular cation HC₃NH⁺: r _1e(CH) = 1.0703Å, R _1e(C₍₁₎C₍₂₎) = 1.2097 Å, R _2e(C₍₂₎C₍₃₎) = 1.3509Å, R _3e(C₍₃₎N) = 1.1448 Å and r _2e(NH) = 1.0079Å. Ground-state rotational constants for less abundant isotopomers are predicted with an uncertainty of about 0.02 MHz. The equilibrium dipole moment of HC₃NH⁺ is calculated to be 1.61 D.     

Line energy,line tension and drop size

The relation between drop radius, r, the force to move the three phase contact line and the advancing and receding contact angles θ_A and θ_R is studied. To keep the line energy (energy per 2πr, also named line tension) independent of r, the modified Young equation predicts that the advancing and receding contact angles, θ_A and θ_R, change considerably with r. As shown by many investigators, θ_A and θ_R change negligibly, if at all, with r. We quantify recent evidences showing that the line energy is a function of the Laplace pressure and show that this way the modified Young equation is correct and still θ_A and θ_R should hardly change with r. According to our model, the small surface deformation associated with the unsatisfied normal component of the Young equation results in higher intermolecular interactions at the three phase contact line which corresponds to a higher retention force. This time increasing effect is supported by recent experiments.     

Atomic Model Calculations of Gain Saturation in the 46.9 nm Line of Ne‐like Ar

The gain saturation in the 46.9 nm line of the Ar⁺⁸ laser is analyzed using an atomic kinetics code. The dependence of the gain (G) on the electron kinetic temperature (T_e) in the region (50 ‐150 eV) is calculated in the quasi steady‐state approximation for the different values of the electron density (N_e) and the plasma radius (r_pl). The influence of radiat on trapping, ion random and mean velocities, Stark line broadening and refraction losses on the gain saturation is taken into consideration. For r_pl = 150‐600 μm, the amplplication (G > 0 cm^‐1) exists in the large temperature/density domain (T_e = 60‐150 eV, N_e = 0.5‐10 × 10¹⁸ cm^‐3). However, the value G_s ∼ 1.4 cm^‐1 required for the gain saturation at the typical plasma length L_pl ∼ 15 cm is reached in the extremely narrow density regions at the high temperatures. The saturation is reached for r_pl = 600 μm at T^s_e = 150 eV in the region N^s_e = 1.8‐2 × 10¹⁸ cm ^‐3, for r_pl = 300 μm at T^s_e = 125 eV and N^s_e = 2.5‐3 × 10¹⁸ cm^‐3, and for r_pl = 150 μm at T^s_e = 110 eV and N^s_e = 3‐4 × 10¹⁸ cm^‐3. The broadest density region (N^s_e = 2 ‐8 × 10¹⁸ cm^‐3) is predicted for the narrowest column (r_pl = 150 μm) at the highest temperature (T^s_e = 150 eV). The operation in the broadest density region N^s_e, should make easier achievement of the gain saturation in the experiments.     

On the Entropy of Schwarzschild Space-Time

In a previous paper by Pollock and Singh, it was proven that the total entropy of de Sitter space-time is equal to zero in the spatially flat case K=0. This result derives from the fundamental property of classical thermodynamics that temperature and volume are not necessarily independent variables in curved space-time, and can be shown to hold for all three spatial curvatures K=0,±1. Here, we extend this approach to Schwarzschild space-time, by constructing a non-vacuum interior space with line element ds ²=e^2λ(r) dt ²?e^?2λ(r) dr ²?r ²(dθ ²+sin² θd? ²), where $\mathrm{e}^{2{\lambda }(r)}=-\frac{1}{2}(1-\frac{r^{2}}{R_{0}^{2}})$ , which matches onto the vacuum exterior Schwarzschild metric in such a way that e^2λ and d(e^2λ )/dr are both continuous at the Schwarzschild radius R ₀=2M. Then we show that the volume entropy is equal to A/4, where $A\equiv 4\pi R_{0}^{2}$ is the area of the apparent horizon, as found by Hawking.