Repulsive term

Here, f/ei is the electionic Hamiltonian including the nuclear-nuclear repulsion terms, Pji is a Caitesian component of the momentum, and Mi the mass of nucleus /. One should note that the bra depends on z while the ket depends on z and that the primed R and P equal their unprimed counterparts and the prime simply denotes that they belong to the bra. 

Rognan published the scoring function FRESNO (fast free energy scoring function), which considers a hydrogen-bond term, a lipophilic terra, a repulsive term for the buried polar surface, a rotational term, and a desolvation terra [82]. 

T he core-core interaction between pairs of nuclei was also changed in MINDO/3 from the fiiriu used in CNDO/2. One way to correct the fundamental problems with CNDO/2 such as Ihe repulsion between two hydrogen atoms (or indeed any neutral molecules) at all di -l.inces is to change the core-core repulsion term from a simple Coulombic expression (/ ., ii = ZaZb/Rab) to ... 

The core-core repulsion terms are also different in MNDO from those in MlNDO/3, with c)l I and NH bonds again being treated separately ... 

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 ... 

Fhe van der Waals and electrostatic interactions between atoms separated by three bonds (i.c. the 1,4 atoms) are often treated differently from other non-bonded interactions. The interaction between such atoms contributes to the rotational barrier about the central bond, in conjunction with the torsional potential. These 1,4 non-bonded interactions are often scaled down by an empirical factor for example, a factor of 2.0 is suggested for both the electrostatic and van der Waals terms in the 1984 AMBER force field (a scale factor of 1/1.2 is used for the electrostatic terms in the 1995 AMBER force field). There are several reasons why one would wish to scale the 1,4 interactions. The error associated wilh the use of an repulsion term (which is too steep compared with the more correct exponential term) would be most significant for 1,4 atoms. In addition, when two 1,4... 

The three-body contribution may also be modelled using a term of the form i ( AB,tAc,J Bc) = i A,B,c exp(-Q AB)exp(-/i Ac)exp(-7 Bc) where K, a, j3 and 7 are constants describing the interaction between the atoms A, B and C. Such a functional form has been used in simulations of ion-water systems, where polarisation alone does not exactly model configurations when there are two water molecules close to an ion [Lybrand and Kollman 1985]. The three-body exchange repulsion term is thus only calculated for ion-water-water trimers when the species are close together. 

We assume that the nuclei are so slow moving relative to electrons that we may regard them as fixed masses. This amounts to separation of the Schroedinger equation into two parts, one for nuclei and one for electrons. We then drop the nuclear kinetic energy operator, but we retain the intemuclear repulsion terms, which we know from the nuclear charges and the intemuclear distances. We retain all terms that involve electrons, including the potential energy terms due to attractive forces between nuclei and electrons and those due to repulsive forces... 

The first three terms in Eq. (10-26), the election kinetic energy, the nucleus-election Coulombic attraction, and the repulsion term between charge distributions at points Ti and V2, are classical terms. All of the quantum effects are included in the exchange-correlation potential... 

The above potential is referred to as a Lennard-Jones or 6-12 potential and is summed over all nonbonded pairs of atoms ij. The first positive term is the short range repulsion and the second negative term is the long range attraction. The parameters of the interaction are Aj and B... The convenient analytical form of the 6-12 potential means that it is often used, although an exponential repulsion term is usually considered to be a more accurate representation of the repulsive forces (as used in MM-t). 

Because of the electron-electron repulsion term in Equation (7.2) the hamiltonian cannot be broken down into a sum of contributions from each electron and the Schrddinger... 

The (dispersion plus repulsion) terms are known as the London or van der Waals forces. Spherical, nonpolar molecules are well described... 

Electrostatic Repulsive Forces. As the distance between two approaching particles decreases, their electrical double layers begin to overlap. As a first approximation, the potential energy of the two overlapping double layers is additive, which is a repulsive term since the process increases total energy. Electrostatic repulsion can also be considered as an osmotic force, due to the compression of ions between particles and the tendency of water to flow in to counteract the increased ion concentration. 

In a force-displacement curve, the tip and sample surfaces are brought close to one another, and interact via an attractive potential. This potential is governed by intermolecular and surface forces [18] and contains both attractive and repulsive terms. How well the shape of the measured force-displacement curve reproduces the true potential depends largely on the cantilever spring constant and tip radius. If the spring constant is very low (typical), the tip will experience a mechanical instability when the interaction force gradient (dF/dD) exceeds the... 

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

The orbital model would be exact were the electron repulsion terms negligible or equal to a constant. Even if they were negligible, we would have to solve an electronic Schrodinger equation appropriate to CioHs " " in order to make progress with the solution of the electronic Schrodinger equation for naphthalene. Every molecular problem would be different. 

Here the integration J dr is over the coordinates of both electrons. Such integrals are therefore eight-dimensional (three spatial variables and one spin variable per electron). Integration over the spin variables is straightforward, but the spatial variables are far from easy a particular source of trouble arises from the electron repulsion term. 

The atomic interactions of the system are derived from a many-body empirical potential, the attractive part of which is expressed within the SMA of the TB theory ", while the repulsive term is a pair-potential of Bom-Mayer type. Accordingly, the total energy of the system is written as ... 

Erep is a pairwise repulsive term which implicitly includes the effect of electron-electron interactions which have been counted twice in Eband nd of ion-ion repulsions . It is usually assumed to be of the Bom-Mayer type / i. 

Thus we find that an explanation of the bonding in H2 and the absence of bonding for He2 lies in the relative magnitudes of attractive and repulsive terms. Quantum mechanics can be put to work with the aid of advanced and difficult mathematics to calculate these quantities, to tell us which is more important. Unfortunately, solving the mathematics presents such an obstacle that only a handful of the very simplest molecules have been treated with high accuracy. Nevertheless, for some time now chemists have been able to decide whether chemical bonds can form without appealing to a digital computer. 

Another problem arises from the presence of higher terms in the multipole expansion of the electrostatic interaction. While theoretical formulas exist for these also, they are even more approximate than those for the dipole-dipole term. Also, there is the uncertainty about the exact form of the repulsive interaction. Quite arbitrarily we shall group the higher multipole terms with the true repulsive interaction and assume that the empirical repulsive term accounts for both. The principal merit of this assumption is simplicity the theoretical and experimental coefficients of the R Q term are compared without adjustment. Since the higher multipole terms are known to be attractive and have been estimated to amount to about 20 per cent of the total attractive potential at the minimum, a rough correction for their possible effect can be made if it is believed that this is a preferable assumption. 

There are two electrons that can both be put into the lower-energy orbital with opposite spins so the electronic energy is 2e+. The internuclear repulsion term must also be included in the total energy expression, giving (through eq. 1.33) ... 

The interaction between two double layers was first considered by Voropaeva et a/.145 These concepts were used to measure the friction between two solids in solution. Friction is proportional to the downward thrust of the upper body upon the lower. However, if their contact is mediated by the electrical double layer associated with each interface, an electric repulsion term diminishes the downward thrust and therefore the net friction. The latter will thus depend on the charge in the diffuse layer. Since this effect is minimum at Eam0, friction will be maximum, and the potential at which this occurs marks the minimum charge on the electrode. 

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