Realistic Potentials

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

In some studies of pure N2 solids more realistic potentials have been used by considering electrostatic interaction in addition. The electrostatic interaction sites are positive charges of q = cj2 = 0.313e at distances 1.044 A away from the molecular center-of-mass on the molecular symmetry axis and negative charges = /4 = —0.313 e at distances 0.874 A respectively [143-145]. 

As stated above, the most important missing piece in protein folding theory is an accurate all-atom potential. Recently there has been much effort in this direction, and much more is needed [48,55,72-77]. The existence of a potential satisfying minimal criteria such as folding and stability for a single protein was demonstrated in [73]. It is not a realistic potential by any means, but its existence validates the all-atom, implicit solvent, Monte Carlo approach as a serious candidate for theory. The method used to derive this potential was ad hoc, and has recently been compared with other standard methods in a rigorous and illuminating study [77]. 

The harmonic approximation is only valid for small deviations of the atoms from their equilibrium positions. The most obvious shortcoming of the harmonic potential is that the bond between two atoms can not break. With physically more realistic potentials, such as the Lennard-Jones or the Morse potential, the energy levels are no longer equally spaced and vibrational transitions with An > 1 are no longer forbidden. Such transitions are called overtones. The overtone of gaseous CO at 4260 cm (slightly less than 2 x 2143 = 4286 cm ) is an example. 

Fig. 2.2. Energy levels for ISW and 3DHO potentials. Each shows major gaps corresponding to closed shells and the numbers in circles give the cumulative number of protons or neutrons allowed by the Pauli principle. In a more realistic potential, the levels for given (n, T) are intermediate between these extremes, in which the lower magic numbers 2, 8 and 20 are already apparent. Adapted from Krane(1987).

Tangney et al.92 studied the friction between an inner and an outer carbon nanotube. Realistic potentials were used for the interactions within each nanotube and LJ potentials were employed to model the dispersive interactions between nanotubes. The intra-tube interaction potentials were varied and for some purposes even increased by a factor of 10 beyond realistic para-meterizations, thus artificially favoring the onset of instabilities and friction. Two geometries were studied, one in which inner and outer tubes were commensurate and one in which they were incommensurate. 

The 6-12 potential is only qualitatively like the realistic potentials that can be derived by calculations at Schroedinger level for, say, Ar-Ar interactions. But it requires careful and detailed study to see how real simple fluids (i.e. one component fluids with monatomic particles) deviate from the behavior calculated from the 6-12 model. Moreover the principal structural features of simple fluids are already quite realistically given by the hard sphere fluid. 

The realistic potential foroff-site recycling/reuse of silver salts and concentrated nitric acid must be evaluated, including recyclers ability to accept, handle, and treat these materials. 

Usually, MD methods are applied to polymer systems in order to obtain short-time properties corresponding to problems where the influence of solvent molecules has to be explicitly included. Then the models are usually atomic representations of both chain and solvent molecules. Realistic potentials for non-bonded interactions between non-bonded atoms should be incorporated. Appropriate methods can be employed to maintain constraints corresponding to fixed bond lengths, bond angles and restricted torsional barriers in the molecules [117]. For atomic models, the simulation time steps are typically of the order of femtoseconds (10 s). However, some simulations have been performed with idealized polymer representations [118], such as Bead and Spring or Bead and Rod models whose units interact through parametric attractive-repulsive potentials. 

The EAM approach appears to provide a formalism within which realistic potentials which describe atomic dynamics can be developed. It should also provide a method for realistically incorporating adsorbates into dynamics simulations. Both of these applications can be considered significant advances, and will help molecular dynamics simulations to continue to contribute to the understanding of technologically important processes. 

It thus follows that for deep levels and small (i.e., close to threshold) the simplified analysis should be valid for a relatively wide range of realistic potentials and wave functions. Note that as long as the matrix elements are indeed reasonably constant for allowed transitions and proportional to for forbidden transitions, it is a good approximation to follow the approach of Grimmeiss et al. (1974) near a band edge, the cross section to this band (a ) is given by... 

The last entry in Table 10.1 is the least well defined of those listed. This is of little importance to us, however, since our interest is in attraction, and the final entry in Table 10.1 always corresponds to repulsion. The reader may recall that so-called hard-sphere models for molecules involve a potential energy of repulsion that sets in and rises vertically when the distance of closest approach of the centers equals the diameter of the spheres. A more realistic potential energy function would have a finite (though steep) slope. An inverse power law with an exponent in the range 9 to 15 meets this requirement. For reasons of mathematical convenience, an inverse 12th-power dependence on the separation is frequently postulated for the repulsion between molecules. 

The kernel A(t/ t>) is a measure of the persistence of As velocity in a collision with initial and final speeds, v and t/, respectively. The expressions for K, A and A are given in [235]. Here, we mention that with a number of simplifying assumptions, such as velocity-independent collision frequencies, etc., the equivalent of Eq. 6.87, can be obtained from Eq. 6.88 computations based directly on Eq. 6.88 and realistic potential and dipole models have never been attempted. 

The use of more realistic potentials, including both repulsive and attractive contributions, results in the more smooth intermediate order curves G(r). As previously, they reveal oscillations thus reflecting the specific law of mutual particle distribution in dense systems. 

Multi dimensional quantum mechanical calculations are needed for the quantitative description of the effects discussed above. Rigorously stated, such calculations are very laborious. In this connection, considerable attention has been paid during the last two decades to the development of simplified methods for resolving the multi-dimensional problems. We refer, for instance, to the method of classic S-matrix [60] and the quantum-mechanical method of the transition state [61]. The advantage of these methods is the use of realistic potential energy surfaces the shortcoming is the fact that only... 

Recently attempts have been made to study systems using yet more realistic potentials and it is likely that considerable progress will be made in this direction in the near future. 

The trajectories are still more complicated if the balls interact with a realistic potential. The right side of Figure 7.1 illustrates a case that is realistic for atoms and molecules, as we discussed in Chapter 3 the interaction potential is attractive at long distances and repulsive at short distances. 

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