Coefficient interrelatedness among polynomial potential functions of diatomic molecules

It is shown that the diatomic potential energy functions of Dunham, SPF and Ogilvie can be easily converted from one another when their coefficients are related. Through Maclaurin expansion and comparison of terms, the coefficients can be related by using the Pascal Triangle. In this paper, the coefficients were related up to the tenth order of δr/R for HX (X = H, Ga, Cl, I). Comparison of all three potential energy curves shows very good agreement for r ≤ 1.5R, thereby verifying the formulated relations. Observation of the plotted potential energy curves for r > 1.5R shows that the difference of the three potential function definitions is not reflected as any consistent trend arising from the related potential functions.     

The generalized potential energy function for diatomic molecules

A new generalized potential energy function is suggested for diatomic molecules. The Dunham, Simons—Parr—Finlan, Thakkar and Ogilvie potentials are shown to be particular cases of the generalized potential energy function. It is also shown that the function suggested may reproduce the path of the potential curve with sufficient accuracy even for the cases of small expansion length.     

Exact solutions and coherent states for a generalized potential

Exact solutions of the vibrational Schrödinger equation for a generalized potential energy function \(\hbox {V(R)}=\hbox {C}_{0}(\mathrm{{R}-\mathrm {R}}_{\mathrm{e}})^{2}/[\hbox {aR}\,+\,(\mathrm{{b}-\mathrm {a}})\hbox {R}_{\mathrm{e}}]^{2}\) are obtained. It includes those of Dunham, Ogilvie and Simons–Parr–Finlan potentials as special cases corresponding to b \(=\) 1, a \(=\) 0, 1/2, 1, respectively. The analytical wave functions derived are useful to test the quality of numerical methods or to perform perturbative or variational calculations for the problems that cannot be solved exactly. Coherent states for generalized potential, which minimize the position–momentum uncertainty relation are also constructed.     

Comparison of spectroscopic potentials and an a priori analytical function. The potential energy curve of the ground state of the sodium dimer, X1Sigmag(+) Na2

The results of a "universal" potential energy function, one that incorporates electronegativity and Slater's effective nuclear charge into a Morse-type function, are compared to spectroscopically derived potential energy curves of the X1Sigmag(+) state of Na2. The function is a priori in that it does not require prior knowledge of the actual potential and has no adjustable parameters. Criteria used to evaluate the performance of the function are comparisons of predicted versus spectroscopic energies at Rydberg-Klein-Rees (RKR) procedure turning points, predicted distances at measured energies versus RKR distances, and eigenvalues derived from the a priori potential versus spectroscopically deduced energy levels. The a priori function describes the Na2 potential with deviations approaching the magnitude of those found among some spectroscopic potentials from different sources. By examining the behavior of the "spectroscopic" parameter of the Morse function, irregularities are found in five of the seven spectroscopic potentials examined. A new procedure is demonstrated for correcting irregularities on the inner branch of spectroscopic potentials at high extents of dissociation and for extending reliably the potential in this region beyond the domain of the measurements.     

The multiscale coarse-graining method. VIII. Multiresolution hierarchical basis functions and basis function selection in the construction of coarse-grained force fields

The multiscale coarse-graining (MS-CG) method is a method for determining the effective potential energy function for a coarse-grained (CG) model of a molecular system using data obtained from molecular dynamics simulation of the corresponding atomically detailed model. The coarse-grained potential obtained using the MS-CG method is a variational approximation for the exact many-body potential of mean force for the coarse-grained sites. Here we propose a new numerical algorithm with noise suppression capabilities and enhanced numerical stability for the solution of the MS-CG variational problem. The new method, which is a variant of the elastic net method [Friedman et al., Ann. Appl. Stat. 1, 302 (2007)], allows us to construct a large basis set, and for each value of a so-called "penalty parameter" the method automatically chooses a subset of the basis that is most important for representing the MS-CG potential. The size of the subset increases as the penalty parameter is decreased. The appropriate value to choose for the penalty parameter is the one that gives a basis set that is large enough to fit the data in the simulation data set without fitting the noise. This procedure provides regularization to mitigate potential numerical problems in the associated linear least squares calculation, and it provides a way to avoid fitting statistical error. We also develop new basis functions that are similar to multiresolution Haar functions and that have the differentiability properties that are appropriate for representing CG potentials. We demonstrate the feasibility of the combined use of the elastic net method and the multiresolution basis functions by performing a variational calculation of the CG potential for a relatively simple system. We develop a method to choose the appropriate value of the penalty parameter to give the optimal basis set. The combined effect of the new basis functions and the regularization provided by the elastic net method opens the possibility of using very large basis sets for complicated CG systems with many interaction potentials without encountering numerical problems in the variational calculation.     

Laser spectroscopy on the A0-X1Σ transition of T1I

The first excited electronic state of TII was investigated by laser excitation and laser fluorescence spectroscopy. Almost completely resolved rotational fine structure was observed with the help of a collimated molecular beam and a single-mode cw dye laser. From the derived Dunham parameters Y_lk a RKR potential for the excited state is calculated. The unusual shape of the potential energy curve can be understood as originating from an avoided crossing of the ionic ground-state potential and a non-bonding covalent state of the atomic asymptote (6p) ²P_1/2(TI)+(5p)^{5 2}P_3/2(I).     

Connection between the Ogilvie and the Murrell–Sorbie potential energy functions

Formulas for relating the parameters of the Murrell–Sorbie and the Ogilvie potentials are developed herein. Unlike the Simons–Parr–Finlan function, the Ogilvie potential is more easily connected with the Murrell–Sorbie potential, as evident from the longer range of agreement. The relationship is useful for generating high order Ogilvie potentials that exhibit the dissociation energy without experimentation, and for using the Murrell–Sorbie parameters in molecular softwares that adopt the Ogilvie function in their algorithms. The relations are invertible so that Ogilvie parameters from spectroscopic data can be applied in molecular softwares that employ Murrell–Sorbie potentials.      

Efficient and reliable numerical integration of exchange-correlation energies and potentials

An adaptive numerical integrator for the exchange-correlation energy and potential is presented. It uses the diagonal elements of the exchange-correlation potential matrix as a grid generating function. The only input parameter is the requested grid tolerance. In combination with a defined cell function the adaptive grid generation scales almost linear with the number of basis functions in a system. With the adaptive numerical integrator the self-consistent field energy error, which is due to the numerical integration of the exchange-correlation energy, converges with increasing adaptive grid size to a reference value. The performance of the adaptive numerical integration is analyzed using molecules with first, second, and third row elements. Especially for transition metal systems the adaptive numerical integrator shows considerably improved performance and reliability.     

Potential energy functions of polyatomic molecules

A direct method is proposed for determining polyatomic potential energy functions, expressed in terms of normal coordinates, which yield a given set of vibrational excitation energies. The method is a modification of the semiclassical technique for computing vibrational energy levels of Percival and Pomphrey. The technique is used to derive potential functions for the NO₂, SO₂ and ClO₂ molecules. With these potentials twenty two higher vibrational excitations energies have been predicted for these molecules and these results differ from the experimental values by at most 3 cm^?1. The computed potential functions are not unique despite the apparent accuracy of the vibrational energy levels. Comparison with the RKR method indicates that the present method must be extended to include rotational perturbations.     

Transition probabilities for the ionization of N2, O2, NO and CO molecules

Franck–Condon factors are presented for the normal and stable isotope-labelled N₂, O₂, NO and CO molecules for transitions to the observed ionized states by using the Rydberg–Klein–Rees (RKR ) potential energy curves of the various electronic states involved. It has been observed that for some transitions, the Franck–Condon factors based on the RKR potential energy curves differ appreciably from those based on the Morse potential function. The effect of isotopic substitution on the transition probabilities is also quite significant in a number of cases.     

The potential energy function for the ground state of CCl calculated from spectroscopic data

The dissociation energy (D_e= 57754±872 cm⁻¹ has been estimated for the ground state of CCl⁺ by fitting a Hulburt-Hirschfelder potential to the RKR turning points. This value of D_e has been used together with molecular constants B_e, ω_e, a_i (i= 1–6) and R_e obtained by Gruebele and co-workers to construct a potential energy function for CCl⁺ in the form of a perturbed Morse oscillator.     

Improved Calculation of Vibrational Energy Levels in F2 Molecule using the RKR Method

The potential energy curves of the ground state X²∑⁺_g of the fluorine molecule have been ac-curately reconstructed employing the Ryderg-Klein-Rees (RKR) method extrapolated by a Hulburt and Hirschfeler potential function for longer internuclear distances. Solving the cor-responding radial one-dimensional Schr?dinger equation of nuclear motion yields 22 bound vibrational levels above v=0. The comparison of these theoretical levels with the experimen-tal data yields a mean absolute deviation of about 7.6 cm^-1 over the 23 levels. The highest vibrational level energy obtained using this method is 13308.16 cm?1 and the relative de-viation compared with the experimental datum of 13408.49 cm^-1 is only 0.74%. The value from our method is much closer and more accurate than the value obtained by the quantum mechanical ab initio method by Bytautas. The reported agreement of the vibrational levels and dissociation energy with experiment is contingent upon the potential energy curve of the F₂ ground state.     

The potential barrier of theB 1Γ u state of Na2

With a semiclassical model the measured energy and width of quasibound levels are described for theB ¹Γ _u state of Na₂. A reliable complete potential function of this electronic state is derived including the potential hump which results from the competition between the electron exchange energy and the long range dipole-dipole interaction. The proposed computation of molecular potentials can be used in the general case and shows some advantages over RKR determinations.     

Improved long range relationship between parameters of the Morse and Rydberg potential functions

Information on parameter relationships between different interatomic potential energy functions is useful when there is a functional mismatch between preferred parameters from one potential function and the adoption of another potential function in computational chemistry softwares. Previous attempts in relating parameters of different potential functions focus on equating the potential curves’ curvatures at the minimum well-depth, which are not accurate for large bond-stretching. In this paper, the long range error is minimized by imposing equal energy integral from equilibrium bond length to bond dissociation. Plotted results for the long range parameter relationship between the Morse and Rydberg potential energy functions reveal excellent agreement for long range interaction.     

Improved Pöschl–Teller potential energy model for diatomic molecules

By employing the dissociation energy and the equilibrium internuclear distance for a diatomic molecule as explicit parameters, we construct an improved Pöschl–Teller potential energy model. We analyze the average absolute deviations of the improved Pöschl–Teller and Morse potentials from the experimental Rydberg–Klein–Rees (RKR) potentials for six diatomic molecules. It is found that the improved Pöschl–Teller potential is more accurate than the Morse potential in fitting experimental RKR potential curves over a large range of internuclear distances for six molecules examined.     

Combining rules for interatomic potential functions of Buckingham form

The geometric mean combining rule for the total interaction energy of two non-bonded atoms, the energy being described by the Buckingham type potential, is suggested. Some numerical examples are given to compare this rule with the combining rules for the exp-six potentials proposed by Mason and Rice.     

CuC、CuN分子基态的结构与分析势能函数

在Cu 的有效核势近似下, 运用密度泛函(B3LYP)方法, 对Cu采用基集合LANL2DZ, 但对其价电子基组的p轨道函数部分做了必要的修改, 而对C、N采用基集合6-311+G(d), 对CuC和CuN分子的微观结构进行了理论计算. 优化并计算了两分子基态的能量, 平衡结构和谐振频率. 根据原子分子反应静力学原理, 导出了CuC和CuN分子基态的合理离解极限和离解能. 应用密度泛函(B3LYP)方法扫描了CuC和CuN分子基态的势能曲线, 并采用最小二乘法拟合了两分子基态的Murrell-Sorbie势能函数及其在平衡位置附近的Dunham展开式. 同时根据Herzberg 和Dunham的公式, 计算了CuC和CuN分子基态的光谱参数.     

Rapidly converging threshold value calculations in screened coulomb potential systems: Critical values of the screening parameter for the Yukawa case

In this work, the threshold values of the screening parameter for Yukawa potential systems are obtained by means of a basis set constructed from Laguerre type functions. The Laguerre basis set is modified by an appropriately chosen extra function in order to imitate the true behaviour of the solutions at the boundary points. The method used is a variational scheme and the numerical results are accurate to thirty decimal points. A scaling parameter is also inserted into the structure of the basis functions, the optimized values of which accelerate the convergence. The main goal of this paper is to develop a method which enables us to calculate the threshold values of the screening parameter for low-lying states. The method is quite general and can be extended to all systems whose potentials decay exponentially when the radial variable goes to infinity.     

Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Within the framework of supersymmetric quantum mechanics method, we study by an algebraic method the arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential by a proper approximation to nonlinear centrifugal term. The explicitly analytical formula of energy levels is derived, and the corresponding bound state wave functions are presented. The function analysis method is used to rederive the same energy levels of the quantum system under consideration to check the validity of this algebraic method. In addition, it is shown from numerical results of energy levels that above certain α parameter depending on the choices of potential parameters V₁ and V₂ the hyperbolical molecular potential cannot trap a particle as it becomes weaker and the energy level starts to turn positive when the potential parameter α becomes larger. © 2014 Wiley Periodicals, Inc.