Ladder operators for the Morse potential

A realization of the raising and lowering operators for the Morse potential is presented. It is shown that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are obtained for the matrix elements of different operators such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied and an approach previously proposed to calculate the Franck–Condon factors is discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Matrix elements for the Morse potential using ladder operators

The algebra of the two-dimensional harmonic oscillator is exploited to obtain matrix elements between eigenstates of the Morse potential. This follows after mapping the latter into the radial equation of the former problem by means of a change of variable and the use of the angular variable as a dummy variable.     

Franck–Condon factors for the Morse potential

The aim of this study is to establish a new representation for the dynamic algebra of the Morse oscillator and to establish the raising and lowering operators based on the properties of the confluent hypergeometric functions. Using the representation we have obtained a recurrent analytic method for the calculus of the Franck–Condon factors. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 655–660, 1997     

Ladder operators for the Kratzer oscillator and the Morse potential

Taking a close look at the Infeld–Hull ladder operators for the Kratzer oscillator system, V(x) = [x² + β(β ? 1)x^?2]/2, we deduce and explicitly construct energy‐raising and ‐lowering operators for the generalized Morse potential system V(z) = (Ae^?4αz ? Be^?2αz)/2, through a canonical transformation that exists between the two systems. For the Morse potential system, we obtain a system of raising and lowering operators P_±(n) (n = 0, 1, 2, 3, … , n_max) with the specific property that P_±(n)Φ_n = c_±(n)Φ_n±1, where Φ_n denotes the nth energy eigenfunction. While P_?(0) annihilates the ground‐state Φ₀, the operator P₊(n_max), instead of annihilating the highest bound‐state Φ, actually knocks it out of the L₂ space spanned by the discrete bound states and becomes inadmissible. Yet, raising and lowering operators _± with proper end‐of‐spectrum behavior (i.e., _?|0〉 = 0 and ₊|n_max〉 = 0) can be constructed in a straightforward way in the energy representation. We show that the operators ₊, _?, and ₀ (where ₀ ≡ (1/2)[ ₊, _?]) form a su(2) algebra only if we restrict them to the (N ? 1)‐dimensional subspace spanned by the lowest (N ? 1) basis vectors, but not in the full (N + 1)‐dimensional space spanned by the discrete bound states [N ≡ n_max ≡ integral part of (1/2)(B/(2α ) ? 1)]. Realization of this su(2) algebra in the position representation (when restricted to the (N ? 1)‐dimensional subspace) is also given. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Modeling intermolecular interactions of physisorbed organic molecules using pair potential calculations

The understanding and control of epitaxial growth of organic thin films is of crucial importance in order to optimize the performance of future electronic devices. In particular, the start of the submonolayer growth plays an important role since it often determines the structure of the first layer and subsequently of the entire molecular film. We have investigated the structure formation of 3,4,9,10-perylene-tetracarboxylic dianhydride and copper-phthalocyanine molecules on Au(111) using pair-potential calculations based on van der Waals and electrostatic intermolecular interactions. The results are compared with the fundamental lateral structures known from experiment and an excellent agreement was found for these weakly interacting systems. Furthermore, the calculations are even suitable for chemisorptive adsorption as demonstrated for copper-phthalocyanine/Cu(111), if the influence of charge transfer between substrate and molecules is known and the corresponding charge redistribution in the molecules can be estimated. The calculations are of general applicability for molecular adsorbate systems which are dominated by electrostatic and van der Waals interaction.     

Model atom approximation for estimating the localized bond energy based on the Morse potential

A method for solving the vibrational problem by using an effective energy operator containing Morse potentials and new model atom approximations is developed. The approach is explained by reference to the trifluoromethylacetylene molecule where the model atom is a CCC or CC atomic group. Using certain assumptions for parameters in the spectrum, we managed to confirm the known experimental result of bond strengthening of the terminal hydrogen atom when the CC bond order increases and to predict the CH bond energy (462.5±4.6 kJ/mole). This procedure seems to be also effective for more complex adsorption problems where the model atom is a crystal. Given that the surface effect is predicted by the parameters of the method, the corresponding model energy operator is constructed. With appropriate spectral data, the approach guarantees very accurate estimations of adsorption bond energies. Translated from Zhumal Struktumoi Khimii, Vol. 38, No. 2, pp. 263–269, March–April, 1997.     

Resonances and antibound states in a Morse potential

We calculate the resonant and antibound state energies for a Morse potential with a centrifugal barrier using Siegert boundary conditions. Starting with a complex wave number k (purely imaginary for bound and antibound states), we integrate numerically from the origin up to a matching point using Numerov's method. The inward integration is performed using the corresponding (first-order) Riccati equation. The complex eigenvalues are found by matching the two logarithmic derivatives. We find narrow shape resonances within the well, above the dissociation limit, and broad resonances above the centrifugal barrier. Antibound states are found even with J = 0. © 1997 John Wiley & Sons, Inc.     

Modification of Morse potential in conventional force fields for applying FPDP parameters

Although the Morse potential function is widely used in molecular modeling software, newer potential functions that possess more parameters provide greater accuracy. Against this backdrop, the Four-Parameter-Diatomic-Potential (FPDP) was selected for converting its parameter into those of the Morse potential due to the former’s resemblance to the latter. A pair of modified Morse indices was extracted by imposing equal force constant for infinitesimal bond stretching and equal energy integral for complete interatomic separation. Results reveal very good agreement for both bond compression and bond stretching. The developed parameter conversion would enable all FPDP parameters to be converted into the modified Morse parameters. Only minor algorithm alterations are required for incorporating the modified Morse function into molecular modeling packages that adopt the conventional Morse potential for describing 2-body bonded interaction.     

New theoretical insight into the interactions and properties of formic acid: development of a quantum-based pair potential for formic acid

We performed ab initio quantum-chemical studies for the development of intra- and intermolecular interaction potentials for formic acid for use in molecular-dynamics simulations of formic acid molecular crystal. The formic acid structures considered in the ab initio studies include both the cis and trans monomers which are the conformers that have been postulated as part of chains constituting liquid and crystal phases under extreme conditions. Although the cis to trans transformation is not energetically favored, the trans isomer was found as a component of stable gas-phase species. Our decomposition scheme for the interaction energy indicates that the hydrogen-bonded complexes are dominated by the Hartree-Fock forces while parallel clusters are stabilized by the electron correlation energy. The calculated three-body and higher interactions are found to be negligible, thus rationalizing the development of an atom-atom pair potential for formic acid based on high-level ab initio calculations of small formic acid clusters. Here we present an atom-atom pair potential that includes both intra- and inter molecular degrees of freedom for formic acid. The newly developed pair potential is used to examine formic acid in the condensed phase via molecular-dynamics simulations. The isothermal compression under hydrostatic pressure obtained from molecular-dynamics simulations is in good agreement with experiment. Further, the calculated equilibrium melting temperature is found to be in good agreement with experiment.     

A proof of the equivalence of two formulas for the matrix elements of the Morse potential

Two important formulas for the matrix elements of the Morse potential are shown to be equivalent. The proof rests on certain theorems relating generalized hypergeometric functions of unit argument. This revised version was published online in July 2006 with corrections to the Cover Date.     

Application of extended-Rydberg parameters in general Morse potential functions

The 3-parameter Rydberg and the 4-parameter general Morse functions have been adopted for describing bond-stretching in computation of molecular systems and many-body solid state systems. However the Extended-Rydberg function, such as the 5-parameter Murrell-Sorbie potential, has so far been confined to describing bond-stretching in very small molecules. Due to the superior functional flexibility of the Murrell-Sorbie potential, these parameters are converted into those of the general Morse potential due to the latter’s application in computational chemistry. Imposition of (a) equal minimum well-depth curvature, and (b) equal equilibrium-to-dissociation energy integral on both functions enable the repulsive and the attractive indices of the general Morse potential to be extracted. Comparison of the Murrell-Sorbie energy of CO, FSi, MgS and ClLi with the converted general Morse energy exhibits very good correlation.     

Morse potential eigen‐energies through the asymptotic iteration method

The eigen‐energies of the rotating Morse potential based on the original Pekeris approximation are obtained by means of the asymptotic iteration method. This approach is applied to several diatomic molecules, and it is shown that the numerical results for the eigen‐energies are all in excellent agreement with the ones obtained before. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

The true diatomic potential as a perturbed Morse function

The problem of representing a diatomic (true) Rydberg-Klein-Rees potential U^t by an analytical function U^a is discussed. The perturbed Morse function is in the form U^a = U^M + ∑b_nyⁿ, where the Morse potential is U^M = Dy², y = 1 ?exp(?;a(r ? r_e)). The problem is reduced to determination of the coefficients b_n so U^a(r) = U^t(r). A standard least-squares method is used, where the number N of b_n is given and the average discrepancy ΔU = |(U^t ? U^a)/U^t| is observed over the useful range of r. N is varied until ΔU is stable. A numerical application to the carbon monoxide X¹∑ state is presented and compared to the results of Huffaker¹ using the same function with N = 9. The comparison shows that the accuracy obtained by Huffaker is reached in one model with N = 5 only and that the best ΔU is obtained for N = 7 with a gain in accuracy. Computation of the vibrational energy E_v and the rotational constant B_v, for both potentials, shows that the present method gives values of ΔE and ΔB that are smaller than those found by Huffaker. The dissociation energy obtained here is 2.3% from the experimental value, which is an improvement over Huffaker's results. Applications to other molecules and other states show similar results. © 1995 by John Wiley & Sons, Inc.     

Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium

The adiabatic, relativistic, and quantum electrodynamics (QED) contributions to the pair potential of helium were computed, fitted separately, and applied, together with the nonrelativistic Born-Oppenheimer (BO) potential, in calculations of thermophysical properties of helium and of the properties of the helium dimer. An analysis of the convergence patterns of the calculations with increasing basis set sizes allowed us to estimate the uncertainties of the total interaction energy to be below 50 ppm for interatomic separations R smaller than 4 bohrs and for the distance R = 5.6 bohrs. For other separations, the relative uncertainties are up to an order of magnitude larger (and obviously still larger near R = 4.8 bohrs where the potential crosses zero) and are dominated by the uncertainties of the nonrelativistic BO component. These estimates also include the contributions from the neglected relativistic and QED terms proportional to the fourth and higher powers of the fine-structure constant α. To obtain such high accuracy, it was necessary to employ explicitly correlated Gaussian expansions containing up to 2400 terms for smaller R (all R in the case of a QED component) and optimized orbital bases up to the cardinal number X = 7 for larger R. Near-exact asymptotic constants were used to describe the large-R behavior of all components. The fitted potential, exhibiting the minimum of -10.996 ± 0.004 K at R = 5.608 0 ± 0.000 1 bohr, was used to determine properties of the very weakly bound (4)He(2) dimer and thermophysical properties of gaseous helium. It is shown that the Casimir-Polder retardation effect, increasing the dimer size by about 2 ? relative to the nonrelativistic BO value, is almost completely accounted for by the inclusion of the Breit-interaction and the Araki-Sucher contributions to the potential, of the order α(2) and α(3), respectively. The remaining retardation effect, of the order of α(4) and higher, is practically negligible for the bound state, but is important for the thermophysical properties of helium. Such properties computed from our potential have uncertainties that are generally significantly smaller (sometimes by nearly two orders of magnitude) than those of the most accurate measurements and can be used to establish new metrology standards based on properties of low-density helium.     

Modified Booth equation for the calculation of zeta potential

 An easy-to-use computer program based on the modified Booth equation (MBE) is developed to calculate the zeta potential of a spherical nonconducting particle from knowledge of the electrophoretic mobility, particle size, and the type and concentration of ions present in the solution. The program is applied to five sets of previously published literature data and the resulting zeta potentials are compared with the values given by the Henry equation to illustrate the extent of the relaxation effect in each case. In four cases, the output zeta potential data are compared with the corresponding values obtained from the rigorous numerical solution of O’Brien and White. Results indicate that the computer program developed here gives a reliable estimate of the zeta potential. The main advantage in using the MBE lies in its capability of calculating the zeta potential even for cases where the experimentally measured mobility exceeds the maximum theoretical mobility predicted by the O’Brien and White solution. Received: 16 June 1997 Accepted: 11 September 1997     

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.     

An accurate analytic representation of the water pair potential

The ab initio water dimer interaction energies obtained from coupled cluster calculations and used in the CC-pol water pair potential (Bukowski et al., Science, 2007, 315, 1249) have been refitted to a site-site form containing eight symmetry-independent sites in each monomer and denoted as CC-pol-8s. Initially, the site-site functions were assumed in a B-spline form, which allowed a precise optimization of the positions of the sites. Next, these functions were assumed in the standard exponential plus inverse powers form. The root mean square error of the CC-pol-8s fit with respect to the 2510 ab initio points is 0.10 kcal mol(-1), compared to 0.42 kcal mol(-1) of the CC-pol fit (0.010 kcal mol(-1) compared to 0.089 kcal mol(-1) for points with negative interaction energies). The energies of the stationary points in the CC-pol-8s potential are considerably more accurate than in the case of CC-pol. The water dimer vibration-rotation-tunneling spectrum predicted by the CC-pol-8s potential agrees substantially and systematically better with experiment than the already very accurate spectrum predicted by CC-pol, while specific features that could not be accurately predicted previously now agree very well with experiment. This shows that the uncertainties of the fit were the largest source of error in the previous predictions and that the present potential sets a new standard of accuracy in investigations of the water dimer.     

New model pair potential for water. Effect of inclusion of specific O⋯H interactions on the topology and dynamics of the hydrogen-bond network

Classical molecular dynamics was used to calculate the topologic and dynamic characteristics of H-bond networks in water, using various model potentials. The explicit inclusion of intermolecular interactions of oxygen and hydrogen atoms leads to a model involving additional stabilization of tetrahedrally coordinated molecules and, as a consequence, increased number of such molecules in the system. The H-bond lifetimes point to a strongly correlated molecular motion in the immediate environment.     

Explicit form of Pauli potential for direct derivation of pair density from a two-particle differential equation for the quintet state of four electrons with harmonic interparticle interactions

Recently, an analytic two-particle density matrix (2DM) has been derived for the quintet state of four electrons interacting via two-body harmonic forces. Here we use this 2DM to extract the exact pair density $\Gamma (\mathbf{r}_1,\mathbf{r}_2)$ . This is then employed in the known two-particle partial differential equation for the pair density amplitude to extract the Pauli potential $v_P(\mathbf{r}_1,\mathbf{r}_2)$ for this quintet state.