Classical Monte Carlo

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

With increasing values of P the molar volume is in progressively better agreement with the experimental values. Upon heating a phase transition takes place from the a phase to an orientationally disordered fee phase at the transition temperature where we find a jump in the molar volume (Fig. 6), the molecular energy, and in the order parameter. The transition temperature of our previous classical Monte Carlo study [290,291] is T = 42.5( 0.3) K, with increasing P, T is shifted to smaller values, and in the quantum limit we obtain = 38( 0.5) K, which represents a reduction of about 11% with respect to the classical value. 

The MD/QM methodology [18] is likely the simplest approach for explicit consideration of quantum effects, and is related to the combination of classical Monte Carlo sampling with quantum mechanics used previously by Coutinho et al. [27] for the treatment of solvent effects in electronic spectra, but with the variation that the MD/QM method applies QM calculations to frames extracted from a classical MD trajectory according to their relative weights. 

Figure 8. Li + H2 Ground-state population as a function of time for a representative initial basis function (solid line) and the average over 25 (different) initial basis functions sampled (using a quasi-classical Monte Carlo procedure) from the Lit2/j) + H2(v — 0, j — 0) initial state at an impact parameter of 2 bohr. Individual nonadiabatic events for each basis function are completed in less than a femtosecond (solid line) and due to the sloped nature of the conical intersection (see Fig. 7), there is considerable up-funneling (i.e., back-transfer) of population from the ground to the excited electronic state. (Figure adapted from Ref. 140.)...

Free Energies of Solvation in Water, in kcal/mole," Obtained by Classical Monte Carlo Simulations Using Two Different Atomic Charge Definitions... 

Fig. 12.9. Comparison of final rotational state distributions of CI2 as obtained from classical Monte Carlo calculations and exact quantum mechanical closecoupling calculations for HeCl2 and NeCl2. Adapted from Halberstadt, Beswick, and Schinke (1991).

The atomic radii may be further refined to improve the agreement between experimental and theoretical solvation free energies. Work on this direction has been done by Luque and Orozco (see [66] and references cited therein) while Barone et al. [67] defined a set of rules to estimate atomic radii. Further discussion on this point can be found in the review by Tomasi and co-workers [15], It must be noted that the parameterization of atomic radii on the basis of a good experiment-theory agreement of solvation energies is problematic because of the difficulty to separate electrostatic and non-electrostatic terms. The comparison of continuum calculations with statistical simulations provides another way to check the validity of cavity definition. A comparison between continuum and classical Monte Carlo simulations was reported by Costa-Cabral et al. [68] in the early 1980s and more recently, molecular dynamics simulations using combined quantum mechanics and molecular mechanics (QM/MM) force-fields have been carried out to analyze the case of water molecule in liquid water [69],... 

Unfortunately, quantitatively reliable quantum chemical calculations of nucleation rates for atmospherically relevant systems would require the application of both high-level electronic structure methods and complicated anharmonic thermochemical analysis to large cluster structures. Such computations are therefore computationally too expensive for currently available computer systems, and will likely remain so for the foreseeable future. Instead, a synthesis of different approaches will probably be necessary. In the future, successful nucleation studies are likely to contain combinations of the best features of both classical (Monte Carlo and molecular dynamics) and quantum chemical methods, with the ultimate objective being a chemically accurate, complete configurational sampling. 

Depending on the shape of the envelope function g t) and the field strengths Fi and F2, the conditions (i) and (ii) may, or may not, be simultaneously fulfilled. Ionization is expected to occur only if both (i) and (ii) are fulfilled. The overlap condition depends essentially on v. Thus, as is swept from small values to large values, overlap can be achieved, and lost again, giving rise to a broad ionization structure. A first qualitative analysis of this structure has already been achieved on the basis of (i) and (ii). The decay to the continuum is approximated by an exponential decay with decay constants determined from a classical Monte Carlo calculation. The decay is assumed to start as soon as (i) and (ii) are fulfilled. On the basis of this model Haifmans et al. (1994) obtained the ionization probabilities as a function of i>i shown as the... 

Fig. 7.8 also shows the results of a classical calculation and a quantum calculation that both confirm the prediction of the giant resonance based on the simple overlap criterion discussed above. The crosses in Fig. 7.8 are the results of classical Monte Carlo calculations. They were performed by choosing 200 different initial conditions in the classical phase space at Iq = 57. The ionization probabihty in this case was defined as the excitation probability of actions beyond the cut-off action Ic = 86. This definition is motivated by experiments that, due to stray fields and the particular experimental procedures, cannot distinguish between excitation above Ic > 86 and true ionization, i.e. excitation to the field-free hydrogen continuum. The crosses in Fig. 7.8 are close to the full line and thus confirm the model prediction. The open squares are the results of quantum calculations within the one-dimensional SSE model. The computations were performed in the simplest way, i.e. no continuum was... 

Fig. 7.8. Ionization probabilities of bichromatically driven hydrogen Rydberg atoms as a function of ui obtained by various methods. Ftdl line resoance overlap model crosses classical Monte Carlo simulations open squares quantum calculations open circles with error bars experimental results. (Adapted from Haffmans et al. (1994).)...

Our ability to understand the structure and properties of water in all its forms has been dramatically enhanced by the use of computer simulation. Early studies of the liquid used simple representations of the potential surface. These were often three or four point-charge distributions, adjusted to fit dipole and quadrupole moments, embedded in a simple spherical nonelectrostatic interaction. The simulations used classical Monte Carlo (MC) or molecular dynamic (MD) calculations, and the water molecules were assumed to be rigid. Recently, more advanced calculations have been based on quantum simulations, the introduction of intramolecular degrees of freedom, and accurate potential surfaces. As one side benefit... 

Figure 20 Left High-resolution transmission electron micrograph image of a single PtMo (3 1) nanoparticle on the edge of a carbon black primary particle (111) and (100) fades are clearly resolved. Right Distribution Pt (light) and Mo (dark) atoms in an fee cubo-octahedral particle containing 1806 Pt atoms and 600 Mo atoms from classical Monte Carlo simulation at 550 K.

Here we demonstrate the degeneracy lifting and appearance of the long-rage orbital order by the classical Monte-Carlo method in a flnite size cluster system [3,7]. We calculate the staggered orbital correlation function... 

Fig. 4 (a) Temperature dependence of the orbital correlation function Mqo, and (b) that of the orbital angle function for several system sizes obtained by the classical Monte-Carlo simulation [3,7]. The system size is given by... 

Now we introduce the numerical results obtained by the classical Monte-Carlo simulation [6]. First we present, in Fig. 14, temperature dependence of the speciflc heat C T) for several system sizes. There is a sharp peak around 0.005 — 0.017 which depends on system size. With increasing a system size, the peak shifts to a lower temperature side and becomes sharp. The peak position is denoted as To from now on. It is worth noting that this value of To is much smaller than the mean-fleld ordering temperature 37/8. To elucidate the orbital state below To, we calculate the... 

Fig. 14 Temperature dependence of the specific heat for several cluster sizes obtained in the classical Monte-Carlo simulation [6]...

As in classical Monte Carlo simulations it turns out that a flat histogram iT(n) of the expansion orders n is not optimal, and again an optimized ensemble can be derived. 

Lattice dynamics calculations on the plastic /3-nitrogen phase are relatively scarce because, obviously, the standard (quasi-) harmonic theory cannot be applied to this phase. Classical Monte Carlo calculations have been made by Gibbons and Klein (1974) and Mandell (1974) on a face-centered cubic (a-nitrogen) lattice of 108 N2 molecules, while Mandell has also studied a 32-molecule system and a system of 96 N2 molecules on a hexagonal close-packed (/3-nitrogen) lattice. Gibbons and Klein used 12-6... 

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