Effective potentials and the Gel''fand—Levitan equation

Moses et al, have derived an algorithm using the Gel'fand-Levitan equation for generating exactly solvable potentials for a particle in a box, harmonic oscillator, and Coulomb potentials by adding or subtracting a finite number of eigenvalues. We propose that their algorithm can be used to evaluate effective potentials for non exactly soluble molecular model Hamiltonians. We show that the algorithm can be used to remove bound states from the spectrum and to obtain an effective potential which supports predissociation resonances only. It can also be used to remove a specific resonance state from the spectrum, and to facilitate evaluations of excited states.     

Designing molecules by optimizing potentials

The astronomical number of accessible discrete chemical structures makes rational molecular design extremely challenging. We formulate the design of molecules with specific tailored properties as performing a continuous optimization in the space of electron-nuclear attraction potentials. The optimization is facilitated by using a linear combination of atomic potentials (LCAP), a general framework that creates a continuous property landscape from an otherwise unlinked set of discrete molecular-property values. A demonstration of this approach is given for the optimization of molecular electronic polarizability and hyperpolarizability. We show that the optimal structures can be determined without enumerating and separately evaluating the characteristics of the combinatorial number of possible structures, a process that would be much slower. The LCAP approach may be used with quantum or classical Hamiltonians, suggesting possible applications to drug design and new materials discovery.     

Sum-rule moment bounds on long-range van der waals potentials

Quantum-mechanical sum rules and results from the theory of moments are employed in the construction of rigorous bounds on the long range van der Waals interactions of atoms and molecules.     

Adiabatic potentials for a bridged three-site electron-transfer system

Born—Oppenheimer potential surfaces for a bridged three-site electron-transfer system are calculated. The adiabatic potentials depend upon both the vibrational sum and difference coordinates. Along the vibrational difference coordinate, the ground-state surface has single-minimun form when the through-bridge electronic coupling is strong, and double-minimum form for weak through-bridge coupling. The results are compared to those previously reported for two-site models.     

Information entropies for eigendensities of homogeneous potentials

For homogeneous potentials, the sum S(T), of position and momentum Shannon information entropies Sr and Sp is shown to be independent of the coupling strength scaling. The other commonly used uncertainty like products also follow similar behavior. The ramifications of this scaling property in the cases of hydrogenlike, harmonic oscillator, Morse, and Poeschl-Teller potentials are discussed with the example of S(T).     

Chemical potential perturbation: extension of the method to lattice sum treatment of intermolecular potentials

The recently developed chemical potential perturbation (CPP) method [S. G. Moore and D. R. Wheeler, J. Chem. Phys. 134, 114514 (2011)] is extended to the lattice (Ewald) sum treatment of intermolecular potentials. The CPP method predicts chemical potentials for a range of composition points using the local (position-dependent) pressure tensor of an inhomogeneous system. When computing the local pressure tensor, one can use the Irving-Kirkwood (IK) or Harasima (H) contours of distributing the pressure. We compare these two contours and show that for a planar interface, the homogeneous pressure and resulting chemical potential can be approximated with the CPP method using either the IK or the H contour, though with the lattice sum method the H contour has much greater computational efficiency. The proposed methods are validated by calculating the chemical potentials of the Lennard-Jones fluid and extended simple point-charge (SPC/E) water, and results show a high level of agreement with respective equations of state.     

Prediction of stability changes upon single-site mutations using database-derived potentials

One of the purposes of studying protein stability changes upon mutations is to get information about the dominating interactions that drive folding and stabilise the native structure. With this in mind, we present a method that predicts folding free-energy variations caused by point mutations using combinations of two types of database-derived potentials, i.e. backbone torsion-angle potentials and distance potentials, describing local and non-local interactions along the chain, respectively. The method is applied to evaluate the folding free-energy changes of 344 single-site mutations introduced in six different proteins and a synthetic peptide. We found that the relative importance of local versus non-local interactions along the chain is essentially a function of the solvent accessibility of the mutated residues. For the subset of totally buried residues, the optimal potential is the sum of a distance potential and a torsion potential weighted by a factor of 0.4. This combination yields a correlation coefficient between measured and computed changes in folding free energy of 0.80. For mutations of partially buried residues, the best potential is the sum of a torsion potential and a distance potential weighted by 0.7. For fully accessible residues, the torsion potentials taken alone perform best, reaching correlation coefficients of 0.87 on all but 10 mutations; the excluded mutations seem to modify the backbone structure or to involve interactions that are atypical for the surface. These results show that the relative weight of non-local interactions along the sequence decreases as the solvent accessibility of the mutated residue increases, and vanishes at the protein surface. On the contrary, the weight of local interactions increases with solvent accessibility. The latter interactions are nevertheless never negligible, even for the most buried residues. Received: 20 May 1998 / Accepted: 3 September 1998 / Published online: 7 December 1998     

Density functional calculation of intermolecular potentials

Calculations of intermolecular potentials following the density functional theory (DFT) turn out to be very complicated without using some appropriate approximations. Most often the following three approximations have been considered. In one approximation the disturbed charge distributions during collisions are reduced to sums of undisturbed charge distributions from the colliding species. In another approximation, the so-called local density approximation (LDA), one neglects the fact that the intermolecular potentials that depend on charge densities also depend on gradients in the densities. In a third approximation one assumes that the intermolecular potential can be considered as a sum of two terms: a term for the long-range geometry and a term for the short-range geometry. In this Article the three approximations mentioned will be discussed for numerical accuracy for calculations of potentials between inert gas atoms and for calculations of potentials between surfaces and inert gas atoms. In the discussion a few other approximations will be mentioned too.     

Interaction potentials of the RG-I anions, neutrals, and cations (RG = He, Ne, Ar)

Interaction potentials of the iodine atom, atomic cation, and anion with light rare-gas atoms from He to Ar are calculated within the unified ab initio approach using the unrestricted coupled-cluster with singles and doubles and perturbative treatment of triples correlation treatment, relativistic small-core pseudopotential, and an extended basis set. Ab initio points are fit to a flexible analytical function. The calculated potentials are compared with available literature data, assessed in the I(-)-and I+-ion mobility calculations and the Ar-I(-)-anion zero electron kinetic-energy spectra simulations, and analyzed using the correlation rules. The results indicate a high precision of the reported potentials.     

Atomic partitioning of two-center potentials for slater basis

In a previous paper we have shown that the long-range potential generated by the two-center distributions can be written as a sum of two multipolar expansions centered at the terminal atoms and defined so that the two series of multipoles be minimal. In this paper we show that both charge distributions and short-range potentials can also be separated in atomic contributions in a way consistent with the above mentioned partition of the long-range potentials.     

Local correlation potentials from Brueckner coupled-cluster theory

Local correlation potentials have been obtained from the nonlocal Brueckner coupled-cluster correlation potentials for the rare-gas atoms He, Ne, and Ar and the CO molecule. It is shown that the local correlation potential can mainly be expressed as a sum of two components: a "pure" correlation part and a relaxation contribution. While the total correlation potentials show an oscillating behavior near the nuclei, indicating the atomic shell structure, their components decrease rather monotonously, with a step structure in case of Ne and Ar. By looking at the determinantal overlap and one-electron properties it has been found that the orbitals obtained from these local potentials form a determinant which very well corresponds with the Brueckner determinant. Thus the previously found closeness between the Hartree-Fock determinant and the exchange-only Kohn-Sham determinant [Della Sala and Gorling, J. Chem. Phys. 115, 5718 (2001)] is confirmed also for the correlated case.     

Charge location and fragmentation under electron impact. V—the ionization potentials of (methylated) phosphoramides,guanidines, formamides,acetamides, ureas and thioureas

The measurement and interpretation of the ionization potentials of some (methylated) phosphoramides, acetamides and formamides is reported. Additivity rules for the prediction of ionization potentials are examined.     

Two-component relativistic methods for the heaviest elements

Different generalized Douglas-Kroll transformed Hamiltonians (DKn, n=1, 2,...,5) proposed recently by Hess et al. are investigated with respect to their performance in calculations of the spin-orbit splittings. The results are compared with those obtained in the exact infinite-order two-component (IOTC) formalism which is fully equivalent to the four-component Dirac approach. This is a comprehensive investigation of the ability of approximate DKn methods to correctly predict the spin-orbit splittings. On comparing the DKn results with the IOTC (Dirac) data one finds that the calculated spin-orbit splittings are systematically improved with the increasing order of the DK approximation. However, even the highest-order approximate two-component DK5 scheme shows certain deficiencies with respect to the treatment of the spin-orbit coupling terms in very heavy systems. The meaning of the removal of the spin-dependent terms in the so-called spin-free (scalar) relativistic methods for many-electron systems is discussed and a computational investigation of the performance of the spin-free DKn and IOTC methods for many-electron Hamiltonians is carried out. It is argued that the spin-free IOTC rather than the Dirac-Coulomb results give the appropriate reference for other spin-free schemes which are based on approximate two-component Hamiltonians. This is illustrated by calculations of spin-free DKn and IOTC total energies, r(-1) expectation values, ionization potentials, and electron affinities of heavy atomic systems.     

Universal method for computation of electrostatic potentials

A computational approach to determine electrostatic interaction and gravitational potentials by performing direct numerical integration is presented. The potential is expanded using finite-element functions of arbitrary order. The method does not involve any solutions of systems of linear equations. The potential is instead obtained as a sum of differential contributions. Thus, no boundary conditions for the potential are needed. It is computationally efficient and well suited for parallel computers, since the innermost loops constitute matrix multiplications and the outer ones can be used as parallel indices. Without using prescreening or other computational tricks to speed up the calculation, the algorithm scales as N4/3 where N denotes the grid size.     

Unique continuation for the magnetic Schrödinger equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.     

Ab initio and analytic intermolecular potentials for Ar-CF4

Ab initio calculations at the CCSD(T) level of theory were performed to characterize the Ar + CF4 intermolecular potential. Potential energy curves were calculated with the aug-cc-pVTZ basis set, and with and without a correction for basis set superposition error (BSSE). Additional calculations were performed with other correlation consistent basis sets to extrapolate the Ar-CF4 potential energy minimum to the complete basis set (CBS) limit. Both the size of the basis set and BSSE have substantial effects on the Ar + CF4 potential. Calculations with the aug-cc-pVTZ basis set, and with a BSSE correction, appear to give a good representation of the BSSE corrected potential at the CBS limit. In addition, MP2 theory is found to give potential energies in very good agreement with those determined by the much higher level CCSD(T) theory. Two model analytic potential energy functions were determined for Ar + CF4. One is a fit to the aug-cc-pVTZ calculations with a BSSE correction. The second was derived by fitting an average BSSE corrected potential, which is an average of the CCSD(T)/aug-cc-pVTZ potentials with and without a BSSE correction. These analytic functions are written as a sum of two-body potentials and excellent fits to the ab initio potentials are obtained by representing each two-body interaction as a Buckingham potential.     

Effects of higher order Jahn-Teller coupling on the nuclear dynamics

In this paper effects of higher order Jahn-Teller coupling terms on the nonadiabatic dynamics are studied. Of particular interest is the case when the potential energy surfaces of the degenerate state show pronounced anharmonicity. In order to demonstrate the effects a two-dimensional E multiply sign in circle e Jahn-Teller model system is treated which is based on the e(') stretching vibration of the photoactive (2)E(') state of NO(3) as a realistic example. The sixth order E multiply sign in circle e Jahn-Teller Hamiltonian is derived in the diabatic representation which is valid for any system with a C(3) rotation axis. This diabatization scheme is compared to lower-order Jahn-Teller Hamiltonians and to symmetry adapted as well as ad hoc approximations. Lower-order potentials result in pronounced quantitative and qualitative differences in the dynamics, including differences in the evolution of mean values, the autocorrelation functions (and thus the corresponding spectra), and the electronic population evolution. In the particular example treated, the results of fourth and fifth order potentials are very similar to the sixth order reference system. In contrast, the approximate sixth order Hamiltonians, though the corresponding adiabatic surfaces seem to be nearly identical, results in pronounced differences. The possible consequences for the dynamics of realistic systems with higher dimensionality are briefly discussed.     

Solvation potentials for flexible chain molecules in solution: on the validity of a pairwise decomposition

The effects of a solvent on the conformation of a flexible n-site solute molecule can be described formally in terms of an n-body solvation potential. Given the practical difficulty in computing such multibody potentials, it is common to carry out a pairwise decomposition in which the n-body potential is approximated by a sum of two-body potentials. Here we investigate the validity of this two-site approximation for short interaction-site chain-in-solvent systems. Using exact expressions for the conformation of an isolated chain, we construct a mapping between the full chain-in-solvent system and its solvation potential representation. We present results for both hard-sphere and square-well systems with n=5 that show that the two-site approximation is sufficient to completely capture the effects of an explicit solvent on chain conformation for a wide range of conditions (which include varying the solvent diameter in the hard-sphere system and varying the chain-solvent coupling in the square-well system). In all cases, a set of two-site potentials (one for each distinct site-site pair) is required. We also show that these two-site solvation potentials can be used to accurately compute a multisite intramolecular correlation function.