Attraction term

In order to calculate higher-order wavefunctions we need to establish the form of the perturbation, f. This is the difference between the real Hamiltonian and the zeroth-order Hamiltonian, Remember that the Slater determinant description, based on an orbital picture of the molecule, is only an approximation. The true Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms ... 

The first step towards the development of appropriate expressions is the decomposition of the nonassociative pair potential into repulsive and attractive terms. In this work we apply the Weeks-Chandler-Andersen separation of interactions [117], according to which the attractive part of the Lennard-Jones potential is defined by... 

The dominant force in ionic liquids is Coulombic attraction between ions. The Coulombic attraction term is given by Equation (3.1-2) ... 

Dropping the attractive term in Equation 24.5 gives the inverse power-law (IPL) repulsive potential [69-72]... 

Eq. (7) is often referred to as the (N,6) Lennard-Jones potential. In particular, (12,6) is popular for mathematical reasons, despite the fact that an exponential form as in Eq. (6) usually describes the repulsive part of the potential better. The potentials shown in Fig. 6.2 form a good description for the physisorbed molecule, but they break down for small distances, where the attractive term in Eq. (6) starts to dominate in an unrealistic way, because for d 0 the repulsive part becomes constant (Vr Cr) while the Van der Waals part continuously goes towards infinity (Vvdw —> ) ... 

We can also understand why gold assumes such a special place among the noble metals with respect to reactivity. If we apply Eq. (80) to O adsorption on Cu, Ag, and Au, the d band is full, and consequently / = 1. As a result, the attractive term in Eq.(80) vanishes and only the repulsive term remains, leading to... 

The orbital distortions due to relativity can be pictured in a qualitative way as follows the relativistic potential appears as a short-range attractive term that, above all, stabilizes s-orbitals and makes them more compact as well as steep close to the nucleus. The same happens to a lesser extent for the p-orbitals. Since the low-lying s- and /7-orbitals are now closer to the nucleus they do shield the nuclear charge better than in the nonrelativistic case. Consequently, d- and /-orbitals are... 

In addition to the static induction effects included in I/scf, the hot Drude oscillators give rise to a 1/r6, temperature-dependent, attractive term. This jkg Ta2/r6 term is the classical thermodynamic equivalent of the London quantum dispersive attraction IEa2/r6. It corresponds to a small perturbation to the London forces, because k T is at least two orders of magnitude smaller than the typical ionization energy IE. The smaller the temperature of the Drude motion, the closer the effective potential is to the SCF potential, making Eq. (9-57) independent of mo, the mass of the oscillators. 

Here the atoms in the system are numbered by i, j, k, l = 1,..., N. The distance between two atoms i, j is ry, q is the (partial) charge on an atom, 6 is the angle defined by the coordinates (i, j, k) of three consecutive atoms, and 4> is the dihedral angle defined by the positions of four consecutive atoms, e0 is the dielectric permittivity of vacuum, n is the dihedral multiplicity. The potential function, as given in equation (6), has many parameters that depend on the atoms involved. The first term accounts for Coulombic interactions. The second term is the Lennard-Jones interaction energy. It is composed of a strongly repulsive term and a van der Waals-like attractive term. The form of the repulsive term is chosen ad hoc and has the function of defining the size of the atom. The Ay coefficients are a function of the van der Waals radii of the... 

V is the Laplacian operator for the th electron vS( is the th electron-nuclear attraction term... 

This type of treatment is particularly interesting as it allows one to understand, in a molecule containing two redox sites, whether or not there is an electronic interaction (independently from its through-bond or through-space nature) between the sites, or, with a more attracting term, an electronic communication between the sites. 

The total stabilization energy of a cluster rarely exceeds 25 kcal mol , i.e., a small fraction of a strong covalent bond energy (ca. 100 kcal mol ). Its partitioning into electrostatic, induction, and dispersion terms differs from cluster to cluster. In some cases, one particular energy term is dominant. More typically, many attractive terms contribute to the overall stabilization of non-covalent clusters, as it often happens to hydrogen-bonded complexes. Nevertheless, the electrostatic interaction plays a dominant role, and in the case of polar subsystems. 

While the structure of clusters responds to directionally specific electrostatic interactions, their stabilization energy reflects also the intervention of less specific dispersion interactions. This is, e.g., the case for stacked DNA base pairs. Stability of these pairs stems from dispersion energy while their structure is determined by dipole-dipole electrostatic interactions. Dispersion energy plays an important role in stabilizing clusters of biomacromolecules, where it may be the dominant attractive term. 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

In the opposite limit of a negative second virial coefficient, V2 < 0, corresponding to the bad or poor solvent regime, the polymer coil will be collapsed due to attraction between monomers. In this case, the attraction term in the free energy is balanced by the third-virial term in a low-density expansion (where we assume that V3 > 0),... 

Their expression for a liquid interaction parameter (L) follows Mott (1968) and starts with the simple sum of an attractive term, Co, and a repulsive term, Cp,... 

In the physisorption process, a gas molecule interacts with several atoms at the surface of the solid. The interaction is composed of two terms an attractive term which diminishes with the distance between the molecule and the surface to the power of —6 and a repulsive term which diminishes with the distance to the power of-12. 

For molecules with more than one electron, precise solutions become even more difficult and time consuming, and additional approximations are sought. The simplest molecule is that of hydrogen, where there are two nuclei A and B, and two electrons 1 and 2. The potential energy of the system is the sum of six electrostatic terms the four attractive terms between A-1, A-2, B-1, and B-2, and the two repulsive terms between A-B and 1-2. We seek solutions to the Schrodinger equation of this hydrogen molecule, and the solution is assumed to be a linear combination of the products of the atomic orbitals, of nucleus A associated with electron 1 multiplied by nucleus B associated with electron 2, plus nucleus A associated with electron 2 multiplied by... 

Let us examine the value of Z under different conditions. The first term is always greater than one, which represents the repulsion term making the volume greater than the ideal gas volume and the second term reduces the value of Z, which represents the attraction term. At a fixed value of T above the critical temperature, compression will cause V to decrease so that Z will drop below one, and further compression will cause V to decrease even more so that Z will rise above one. When the temperature is at or below the critical temperature, compression will eventually cause the gas to condense into a liquid at or above the critical pressure Pc. The relations between the critical constants and the values of van der Waals a and b are... 

Develop correlations for the van der Waals attraction term a and the volume term b, using the molecular weight as the predictor. 

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