Induced Dipole Models

Financial support from the NIH, GM51501 to ADM Jr., and GM072558 to BR and ADM Jr. is acknowledged. 

Ponder JW, Case DA (2003) Force fields for protein simulations. In Daggett V, Eisendberg DS, Richards FM, Kuriyan J (eds) Protein Simulations, vol 66. Elsevier Academic Press, New York, 

Mackerell AD (2004) Empirical force fields for biological macromolecules Overview and issues. J Comput Chem 25(13) 1584-1604 

Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79(2) 926 

Cornell WD, Cieplak P, Bayly Cl, Gould IR, Merz KM, Ferguson DM, Spellmeyer DC, Fox T, Caldwell JW, Kollman PA (1995) A 2nd generation force-field for the simulation of proteins, nucleic-acids, and organic-molecules. J Am Chem Soc 117(19) 5179—5197 

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Pulsed source techniques have been used to study thermal energy ion-molecule reactions. For most of the proton and H atom transfer reactions studied k thermal) /k 10.5 volts /cm.) is approximately unity in apparent agreement with predictions from the simple ion-induced dipole model. However, the rate constants calculated on this basis are considerably higher than the experimental rate constants indicating reaction channels other than the atom transfer process. Thus, in some cases at least, the relationship of k thermal) to k 10.5 volts/cm.) may be determined by the variation of the relative importance of the atom transfer process with ion energy rather than by the interaction potential between the ion and the neutral. For most of the condensation ion-molecule reactions studied k thermal) is considerably greater than k 10.5 volts/cm.). 

The results in Table IV suggest that the condensation reactions cannot be described adequately by the ion-induced dipole model. In this regard the results agree with conventional studies which have frequently found a higher power inverse dependence of the cross-section on the field strength E for condensation reactions than for hydrogen transfer reactions. 

Thole s polarizability parameters were selected to optimize the molecular polarizabilities for a set of 16 molecules. The method was later expanded to fit 52 molecules [146], It must be emphasized that this electric-field damping method is totally independent of the polarization scheme used. For the Drude and fluctuating charge methods only /i(r) is required, whereas for methods based on induced dipoles both /i(r) and /2(r) are necessary. In the context of the induced dipole model other models were proposed since the formula of Thole does not provide enough attenuation. For example, in Ref. [152] the field is evaluated using... 

When the Drude particles are treated adiabatically, a SCF method must be used to solve for the displacements of the Drude particle, d, similarly to the dipoles Jtj in the induced dipole model. The implementation of the SCF condition corresponding to the Born-Oppenheimer approximation is straightforward and the real forces acting on the nuclei must be determined after the Drude particles have attained the energy minimum for a particular nuclear configuration. In the case of N polarizable atoms with positions r, the relaxed Drude particle positions r + d5CF are found by solving... 

This equation is the equivalent of Eq. (9-12) for the induced dipole model but has one important difference. Equation (9-13), the derivative of Eq. (9-12), is linear and standard matrix methods can be used to solve for the p. because Eq. (9-12) is a quadratic function of p , while Eq. (9-54) is not a quadratic function of d and thus matrix methods are usually not used to find the Drude particle displacements that minimize the energy. 

An important alternative to SCF is to extend the Lagrangian of the system to consider dipoles as additional dynamical degrees of freedom as discussed above for the induced dipole model. In the Drude model the additional degrees of freedom are the positions of the moving Drude particles. All Drude particles are assigned a small mass mo,i, taken from the atomic masses, m, of their parent atoms and both the motions of atoms and Drude particles (at positions r, and rdj = r, + d, ) are propagated... 

If, for example, the induced dipole model is truncated at the order R 6 in the separation R between the molecular centers, account may be made of the dipoles induced by multipoles up to order — 4 (hexadecapole). Moreover, dipoles induced by derivatives of the local field at their center... 

Spectroscopic measurement. Specifically, if the induced dipole moment and interaction potential are known as functions of the intermolecular separation, molecular orientations, vibrational excitations, etc., an absorption spectrum can in principle be computed potential and dipole surface determine the spectra. With some caution, one may also turn this argument around and argue that the knowledge of the spectra and the interaction potential defines an induced dipole function. While direct inversion procedures for the purpose may be possible, none are presently known and the empirical induced dipole models usually assume an analytical function like Eqs. 4.1 and 4.3, or combinations of Eqs. 4.1 through 4.3, with parameters po, J o, <32, etc., to be chosen such that certain measured spectral moments or profiles are reproduced computationally. 

If one now chooses the appropriate induced dipole model, Eqs. 4.1 through 4.3, or a suitable combination of these, with N parameters po, >7, R0,. .., and one has at least N theoretical moment expressions available, an empirical dipole moment may be obtained which satisfies the conditions exactly, or in a least-mean-squares fashion [317, 38]. We note that a formula was given elsewhere that permits the determination of the range parameter, 1/a, directly from a ratio of first and zeroth moments it was used to determine a number of range parameters from a wide selection of measured moments [189]. In early work, an empirical relationship between the range parameter and the root, a, of the potential is assumed, like 1/a 0.11 a. That relationship is, however, generally not consistent with recent data believed to-be reliable. 

Table 4.1. Empirical induced dipole models for the translational band Eq. 4.30, with n = 1 where B(n) 0.

The last column of Table 4.3 or, equivalently, Eq. 4.30 with the parameters as specified, probably represent the best induced dipole model for He-Ar pairs currently available. This model permits a close reproduction of the measured binary spectra of helium-argon mixtures in the far infrared, see Fig. 5.5 on p. 243. 

The question raised above was is the principle of corresponding states applied to induced dipole surfaces (perhaps approximately) valid If the dipoles were well represented by an exponential function, like Eq. 4.1, the principle would of course be valid. However, it is quite clear that the dispersion part cannot be neglected in the induced dipole models. Moreover, for the best dipole models presently available, we need nonvanishing values b, Eq. 4.30, see p. 162. In other words, the induced dipole surfaces of rare gas pairs seem to require four-parameter functions, like Eq. 4.30, and it is not at all clear how these four parameters may be reduced to but two. The principle, therefore, seems at present to be of little practical value (if it were valid at all). 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

Early work. Previous numerical estimates [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in... 

Previous work of the kind was generally based on empirical induced dipole models whose parameters were adjusted to fit measured spectra. For molecular systems like hydrogen pairs, empirical dipole models are highly simplified, for example, by either suppressing the anisotropic overlap terms, the AL = 21 components, in favor of an overlap term in the... 

Here E is the total electric field at the position of the Drude particle, r - d, arising from the fixed charges as well as all the induced dipoles (modeled with Drude oscillators). For atomic positions r,the relaxed displacements produce the potential... 

The Drude oscillator model has a number of advantages over other polarizable models facilitating its implementation in multiple simulation packages including CHARMM [150], NAMD [165], ChemSell QM/MM [192] and the OpenMM suite of utilities for GPU [193]. Representing a dipole as two point charges provides an intuitive physical picture in terms of displacement of the electronic distribution the model is able to represent delocalization without need of additional non-atomic sites since the dipole is not point-like as, e.g., in the induced dipole model. For example, the use of auxiliary particles allows for the inclusion of mechanical polarizabilities [194]... 

Fig. 20. Comparison between a model potential energy curve (representing the hard-sphere plus ion-induced-dipole model), shown as a dashed line, and a realistic potential curve from a Hartree-Fock calculation, shown as a solid line. The Hartree-Fock result is for the state of ArJ, as illustrated in Figure 21.

WangJ, Cieplak P, Li J, et al. Development of polarizable models for molecular mechanical calculations II induced dipole models significantly improve accuracy of inter-molecular interaction energies. J Phys Chem B. 2011 115(12) 3100—3111. http // dx.doi.org/10.1021/jpll21382. 

Higher-order multipole moments enhance the forces between particles at short distances and their neglect is extremely questionable, especially if fine effects are looked at, as for instance the ground-state properties of close-packed lattice structures [244,246-251] or the viscosity To go beyond the point dipole approximation Klingenberg and co-workers [ 173,252] developed an empirical force expression for the interaction between two dielectric spheres in a uniform external field from the munerical solution of Laplace s equation [253]. Recently, Yu and co-workers [254,255] proposed a computationally efficient (approximate) dipole-induced-dipole model based on a multiple image method which accounts partially for multipolar interactions. 

MODELING POLARIZATION EFFECTS Induced Dipole Models... 

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