The induction term

The second term of the interaction energy, IND, is always negative. IND is related to the mutual polarization of the electronic charge distributions of A and B (the nuclei are held fixed), inducing additional stabilizing effects. This induction (or polarization) energy contribution is defined in a different way in variational and perturbation theory approaches. Perturbation theory approaches are compelled to compute at the first order of the perturbation scheme only the effects due to the polarization of A with respect to the B distribution kept fixed, and in parallel the effects due to the polarization of B with A kept fixed. Mutual induction effects are introduced at higher order of the perturbation theory and have to be separated in some way from dispersion effects computed at the same time. 

In the variational approach use is made of an extension of the simple technique we have used for ES. 

The separation between electrons of A and B is maintained but the product is now subjected to a constrained variational optimization using the Hamiltonian Hab- The two wave functions are so changed, allowing the effects of mutual polarization, because of the presence of the Vab term in the Hamiltonian they will be so indicated as I a and f g.  

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Table 4.4.1 lists typical values of dipole moments and polarizabilities of some simple molecules and the three coefficients of the r 6 term at 300 K. Except for H2O which is small and highly polar, the dispersion term dominates the long-range energy. The induction term is always the least significant. 

Equation (1-180) can be obtained from a variational calculation of the energy of the dimer with a wave function that is product of determinantal wave functions of the monomers constructed from the optimal (selfconsistent) orbitals. Unfortunately, the optimization of the orbitals of one monomer in the electrostatic field of the other does not prevent the unphysical transfer of electrons from one system to the other, since the antisymmetry of the wave function of the dimer is not preserved, and consequently, the Pauli principle is not satisfied. This may lead to some unphysical results in the short range260. It should also be stressed that the interaction part of Eq. (1-180) cannot be obtained, as proposed in Ref. (261), by taking the expectation value of the interaction operator V with the product of determinantal wave functions of the monomers constructed from the optimal (selfconsistent) orbitals. This would result in an overcounting of the induction terms, already in the second order. 

The induction term is calculated using atomic and bond anisotropic polarizabilities to reproduce molecular polarizability, but fitted with isotropic atomic polarizabilities. 

As a consequence of averaging, the most important contribution to intennolecular energy comes from the dispersion term for all but very polar molecules. Classic examples to illustrate this are given by Davies p. 166). The dispersion term is preponderant for CO and HCl, the dispersion and electrostatic terms are similar for NH3 and the electrostatic contribution is the largest for HjO. The induction term is relatively small in all cases. 

A new empirical potential for water has been developed using spectroscopic data, which is able to model condensed water with good accuracy.483 The potential is referred to as the VRT(ASP-W)III potential (the third fitting of the Anisotropic Site Potential with Woemer dispersion to Vibration-Rotation Tunnelling data). It give excellent results for vibrational properties of water clusters up to (H20)6, but unlike earlier spectroscopically derived potentials also models the liquid state well. MC simulations are used to study the liquid state properties. It is noted that this potential only partly accounts for many-body interactions (the induction term) and the simulations do not include... 

Figure 3.6 illustrates an A-SAPT analysis of the polarization of a benzene by a Na+ cation in the same plane. This is an attractive interaction that stabilizes the complex. The darker colored regions of the benzene represent portions of the molecule that are more important to the induction term in a SAPTO computation. Prior to the A-SAPT analysis, we had expected that polarization of the tt-cloud might be the primary contributor to the large, stabilizing induction energy. Instead, A-SAPT demonstrates that while C-C 7r-electrons are strong contributors, there are also important contributions from nearby C-C a and C-H a bonds. 

Figure 3.6 A-SAPTO/jun-cc-pVDZ voxel visualization for the induction term involving the polarization of benzene by a Na" cation in the same plane. Darkly shaded areas correspond to strong contributions to the attractive induction energy.

The interaction energy of N molecules is not pairwise additive i.e., it is not the sum of the interactions of all possible pairs of molecules. Among the energy corrections up to the second order, the exchange and, first of all, the induction terms contribute to the non-additivity. The electrostatic and dispersion (in the second order) contributions ate pairwise additive. 

It is possible to define a molecular index P for the induction term to be used in combination with the MEP Va to get a detailed description of the spatial propensity of the molecule to develop electrostatic interactions of classical type. Both functions are used under the form of an interaction with a unit point charge q placed at position r. In the case of Va this means a simple multiplication in the case of Pa there is the need of making additional calculations (to polarize the charge distribution of A). There are fast methods to do it, both at the variational level and at the PT level. The analysis of Pa has not yet extensively been used to model IND contributions to AE, and it shall not be used here. This remark has been added to signal that when one needs to develop interactions potentials for molecular not yet studied interactions including, e.g., complex solutes, the use of this approach could be of considerable help. 

Equations 16 and 23 [or (1.15) and (1.22)] can be used to simulate either a segment of a blood vessel or the entire blood vessel itself. In small blood vessels, the inductance L is very low compared to the resistance term R, and therefore the inductance term can be neglected in small arteries, arterioles, and capillaries. Since there is no oscillation of pressure in the capillaries, the inductance term can be neglected in vessels downstream of the capillary (i.e., venules, veins, and vena cava, etc.). 

The equilibrium of the system composed of the solute and the continuum implies that free energy is minimum, fhe main terms which arc assumed to vary during the solvation process being the electrostatic and the induction terms, the geometry and the electron distribution of the solute wdl be modified in order to minimize the sum of their contributions to the free energy of the system. 

The interaction energy at long range can be separated into a number of distinct contributions, with characteristics shown in Table 1. The electrostatic term is a straightforward classical interaction between the charge distributions of the two molecules, while the induction terms arise from the perturbation of each molecule by the electrostatic field of the other. The dispersion energy arises from correlated fluctuations in the charge... 

The analytical calculations in the framework of the long-range approximation can be carried out following the Sect. 4.1.2. Then, for the CH4-N2 complex, using the symmetry properties of the molecules CH4 (A) and N2 (B), the induction term in Eq. (4.2.15) takes the form... 

A realistic model must account not only for the classical electrostatic interaction Ugi, but also for three interactions accounting for the quantum nature of electrons. The exchange-repulsion, or van der Waals repulsion U gp is a consequence of the Pauli principle, while the dispersion (van der Waals attraction) arises from correlated fluctuations of the electrons. Last, the induction term reflects the distortion of the electron density in response to electric fields, including incipient charge transfer associated with bond formation. In molten salts, all these interactions can be taken into account in molecular dynamics (MD) simulations in the framework of the polarizable ion model [1],... 

The SPC model does not include an induction energy term. However, a revised version of this model, SPC/E, is based on a mean-field form (see earlier section on Electrostatics) for the induction term in Eq. [1]. Parameter changes resulted in a 3% increase of the charges. The revised potential generates a more stable hydrogen bond network than does SPC, as well as a self-diffusion coefficient that is in good agreement with experiment. The inclusion of the mean-field correction is also used by two other effective potentials, the and the reduced effective representation (RER) potentials. 

For many systems, the induction term is the dominant many-body term in the interaction energy. However, dispersion and exchange-dispersion have non-zero three-body contributions, which are sometimes added in the force fields explicitly [70-73]. 

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