Induced dipole computation

Polarization is usually accounted for by computing the interaction between induced dipoles. The induced dipole is computed by multiplying the atomic polarizability by the electric field present at that nucleus. The electric field used is often only that due to the charges of the other region of the system. In a few calculations, the MM charges have been included in the orbital-based calculation itself as an interaction with point charges. 

Eo and E (Afi(i)) are respectively the electric fields generated by the permanent and induced multipoles moments. a(i) represents the polarisability tensor and Afi(i) is the induced dipole at a center i. This computation is performed iteratively, as Epoi generally converges in 5-6 iterations. It is important to note that in order to avoid problems at the short-range, the so-called polarization catastrophe, it is necessary to reduce the polarization energy when two centers are at close contact distance. In SIBFA, the electric fields equations are dressed by a Gaussian function reducing their value to avoid such problems. 

Calculation of the energy and forces acting on a molecular system requires knowledge of the magnitude of the inducible dipoles. The forces associated with the dipoles (spatial derivatives of the potential) [13], can be computed from Eq. (9-12), and on atomic site k are... 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

Ma BY, Lii JH, Allinger NL (2000) Molecular polarizabilities and induced dipole moments in molecular mechanics. J Comput Chem 21(10) 813—825... 

MM3(2000) has also included a new approach to obtain bond polarizabilities and induced dipole moments.69 A general formula based on the original MM3 force constants and bond polarizabilities was derived and is used to compute bond polarizabilities, and then molecular polarizabilities by an additive model. 

As a demonstration of the power and versatility of the MM3(2000) force field, a comparative study of dipole moments was computed on forty-four small organic molecules. A segment of those results are discussed here with an emphasis on the improvement of the MM3(2000) force field due to the inclusion of the induced dipole moments. 

Note that since both /z and E are vector quantities, a is a second-rank tensor. The elements of a can be computed through differentiation of Eqs. (9.1) and (9.2). The difference between die permanent electric dipole moment and that measured in the presence of an electric field is referred to as the induced dipole moment. 

Induced dipoles of other pairs have also been obtained by quantum chemical computations [44], Whereas these computations are not as sophisticated as the ones mentioned above and close agreement with observations is not achieved for some of the systems considered, in the case of Ne-Ar they have resulted in a dipole surface that reproduces the best absorption measurements closely. The Ne-Ar induced dipole may, therefore, be recommended as a reliable, but perhaps semi-empirical surface (because its reliability is judged not solely on theoretical grounds). Spectral moments computed with that surface are also given in Table 3.1. 

Similar rototranslational spectra (which may be roughly approximated by the envelope of their stick spectra) are observed in other gases as well. Figures 3.22 and 3.23 show the binary absorption spectra of pure methane and carbon dioxide. The smooth curves drawn through the data points represent line shape computations based on the multipole-induction model of the induced dipoles involved [75, 56, 141, 186]. A detailed analysis indicates that for the CH4-X system, CH4 octopole and hexadecapole induction both contribute roughly in comparable amounts to the observable spectra. Rototranslational spectra of several other systems are known see, for example, a review [58]. 

Here, the pt are the permanent dipoles of molecules i = 1 and 2, and the ptj( r, i 2, Rij) are the dipoles induced by molecule i in molecule j the are the vectors pointing from the center of molecule i to the center of molecule j and the r, are the (intramolecular) vibrational coordinates. In general, these dipoles are given in the adiabatic approximation where electronic and nuclear wavefunctions appear as factors of the total wavefunction, 0(rf r) ( ). Dipole operators pop are defined as usual so that their expectation values shown above can be computed from the wavefunctions. For the induced dipole component, the dipole operator is defined with respect to the center of mass of the pair so that the induced dipole moments py do not depend on the center of mass coordinates. For bigger systems the total dipole moment may be expressed in the form of a simple generalization of Eq. 4.4. In general, the molecules will be assumed to be in a electronic ground state which is chemically inert. 

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. 

Highly developed quantum chemical methods exist to compute with an ever increasing precision molecular and supermolecular properties from first principles. For example, attempts to compute intermolecular interaction potentials and, more recently, induced dipole moments, are well known for the simpler atomic and molecular systems. 

The comparison of spectral line shapes computed on the basis of the ab initio dipole surface of He-Ar with absorption measurements has demonstrated the soundness of the data. The agreement indicates that exchange effects due to intra-atomic correlation and higher-order dispersion terms contribute significantly to the induced dipole. However,... 

Recent work improved earlier results and considered the effects of electron correlation and vibrational averaging [278], Especially the effects of intra-atomic correlation, which were seen to be significant for rare-gas pairs, have been studied for H2-He pairs and compared with interatomic electron correlation the contributions due to intra- and interatomic correlation are of opposite sign. Localized SCF orbitals were used again to reduce the basis set superposition error. Special care was taken to assure that the supermolecular wavefunctions separate correctly for R —> oo into a product of correlated H2 wavefunctions, and a correlated as well as polarized He wavefunction. At the Cl level, all atomic and molecular properties (polarizability, quadrupole moment) were found to be in agreement with the accurate values to within 1%. Various extensions of the basis set have resulted in variations of the induced dipole moment of less than 1% [279], Table 4.5 shows the computed dipole components, px, pz, as functions of separation, R, orientation (0°, 90°, 45° relative to the internuclear axis), and three vibrational spacings r, in 10-6 a.u. of dipole strength [279]. 

Figure 4.1 shows the four significant induced dipole components for the rototranslational bands (left panel). The isotropic and anisotropic overlap components, B01 and — B21, dominate at near range (dotted). These fall off roughly exponentially with separation R so that at more distant range, the quadrupole-induction, B23, dominates it falls off more slowly, like R 4. A weak hexadecapole component, B45, is also present. The dashed lines show the classical (i.e., overlap-free) multipole induction contributions. These differ only at near range from the computed B23 and B45 components,... 

H2-H2 dipole. Early attempts to calculate the induced dipole moments from first principles were described elsewhere [281]. Only in recent times could the substantial problems of such computations be controlled and precise data be generated by SCF and Cl calculations, so that that the basis set superposition errors were small enough and the Cl excitation level is adequate for the long-range effects. The details of the computations are given elsewhere [282, 281],... 

In recent molecular dynamics studies attempts were made to reproduce the shapes of the intercollisional dip from reliable pair dipole models and pair potentials [301], The shape and relative amplitude of the intercollisional dip are known to depend sensitively on the details of the intermolecular interactions, and especially on the dipole function. For a number of very dense ( 1000 amagat) rare gas mixtures spectral profiles were obtained by molecular dynamics simulation that differed significantly from the observed dips. In particular, the computed amplitudes were never of sufficient magnitude. This fact is considered compelling evidence for the presence of irreducible many-body effects, presumably mainly of the induced dipole function. 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Comparison with measurement. Measurements of the absorption of rare gas mixtures exist for some time. This fact has stimulated a good deal of theoretical research. A number of ab initio computations of the induced dipole moment of He-Ar are known, including an advanced treatment which accounts for configuration interaction to a high degree see Chapter 4 for details. Figure 5.5 shows the spectral density profile computed... 

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