Evaluation of interaction energies

Interaction energy between molecules A and B (AE ) is determined as the difference between the energy of the dimer (E ) and the sum of the monomer 

The methodology is extended accordingly for tetramers and larger complexes. Analysis of extended systems can be simplified by treating a selected group of molecules as one subsystem. For example a hydrated cation (cation plus water molecules) could be treated as one subsystem when evaluating the nonadditivity of interactions in complexes between hydrated cations and base pairs and trimers [13c]. 

The stability of a dimer is further influenced by the deformation of the monomers upon formation of the complex. This can be evaluated by subtracting the energy of the optimized isolated monomers (E, E ) from the energies of the monomers in the dimer geometry (E, E ). The respective deformation (relaxation) energy AE is positive (repulsive). 

If the calculations are made with inclusion of electron correlation effects, then interaction energies AE and their components consist of HF and electron-correlation components 

The former term basically includes the electrostatic, induction, charge-transfer, and electron-exchange contributions. The dispersion energy originates in the electron correlation which also influences the other contributions. 

---

A few groups replace the Lennard-Jones interactions by interactions of a different form, mostly ones with a much shorter interaction range [144,146]. Since most of the computation time in an off-lattice simulation is usually spent on the evaluation of interaction energies, such a measure can speed up the algorithm considerably. For example, Viduna et al. use a potential in which the interaction range can be tuned... 

The study of composite cations encounters further problems for classical and conventional QM/MM simulations, as their lower symmetry makes the evaluation of interaction energy surfaces and analytical potential functions describing them difficult. In these cases the QMCF MD method provides an elegant solution as well, renouncing solute-solvent potential functions. This advantage could be well demonstrated in studies on the dimer of Hg(I) (39), the titanyl ion (64), and the uranyl ions of U(V) (65) and U(VI) (66). Whereas the Hg + ion still has a fairly regular hydration structure although with a quite peculiar shape, the... 

Various other simplified methods of treating electron spin have been developed, such as those of Pauli,1 Darwin,2 and Dirac.3 These are especially useful in the approximate evaluation of interaction energies involving electron spins in systems containing more than one electron. 

Evaluation of electric multipole moments (and also electrostatic and HF interaction energies) does not pose a problem and double-zeta (DZ) quality basis sets with one set of d-polarization functions usually give proper values. A correct description of the polarizability and hence also the dispersion (intersystem correlation) energy requires much larger basis sets. The presence of polarization functions is inevitable and these functions must be diffuse. This means that dispersion-optimized polarization functions must be used instead of energy-optimized ones for evaluation of interaction energies using medium-sized basis sets. 

For quantitative evaluation of ERDA energy spectra considerable deviations of recoil cross-sections from the Rutherford cross-section (Eq. 3.51) must be taken into account. Light projectiles with high energy can penetrate the Coulomb barrier of the recoil atom the nuclear interaction generally leads to a cross-section that is larger than ctr, see Eq. (3.51). For example, the H recoil cross-section for MeV He projec-... 

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

In this approach we give quantitative and semi-quantitative evaluation of spatial-energy interactions at main stages of complicated biophysical process of photosynthesis based on the utilization of initial atomic characteristics. The analysis of results after the application of P-parameter methodology shows that they correspond to reference data both in the direction and energetics of these processes. 

We have already introduced the concept of ionic polarizability (section 1.8) and discussed to some extent the nature of dispersive potential as a function of the individual ionic polarizability of interacting ions (section 1.11.3). We will now treat another type of polarization effect that is important in evaluation of defect energies (chapter 4). 

Cruciani et al., used a dynamic physicochemical interaction model to evaluate the interaction energies between a water probe and the hydrophilic and hydrophobic regions of the solute with the GRID force field. The VolSurf program was used to generate a PLS model able to predict log Poet [51] from the 3D molecular structure. 

One way to make this process somewhat more efficient is to adopt rigid structures for the various molecules. Thus, one does not attempt to perform geometry optimizations, but simply puts the molecules in some sort of contact and evaluates their interaction energies. To that extent, one needs only to evaluate non-bonded terms in the force field, like those... 

The calculation of molecular properties is as important as the evaluation of the energy, and a recent paper on spin-rotation interaction and magnetic shielding in molecules, using the Kolos-Wolniewicz (KW) wavefunction,40 shows that theory and experiment are in good agreement, if an old approximation in the spin-rotation interaction theory is removed.55... 

Even for overlap integrals as large as 0.3, the second term in equation (97) is of the order 0. and the third term 0.01 Afk, so that except for very close interactions, this series may be safely cut off after the third term. This may be useful for evaluating the interaction energy between the closed-shell atomic cores within a molecule, for which just the first term may well be sufficient. 

Szczesniak, M. M. and Scheiner, S., Accurate evaluation of SCF and MP2 components of interaction energies. Complexes ofHF, OH, and NH with Li, Coll. Czech. Chem. Commun. 53, 2214-2229 (1988). 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

---