Monte Carlo methods density functional theory

A good starting place for information about Monte Carlo and density functional methods is Wikipedia http jjen. wikipedia. org/wiki/Monte Carlo method. http I I en. wikipedia. org/wiki / Density functional theory. 

Ab initio methods allow the nature of active sites to be elucidated and the influence of supports or solvents on the catalytic kinetics to be predicted. Neurock and coworkers have successfully coupled theory with atomic-scale simulations and have tracked the molecular transformations that occur over different surfaces to assess their catalytic activity and selectivity [95-98]. Relevant examples are the Pt-catalyzed NO decomposition and methanol oxidation. In case of NO decomposition, density functional theory calculations and kinetic Monte Carlo simulations substantially helped to optimize the composition of the nanocatalyst by alloying Pt with Au and creating a specific structure of the PtgAu7 particles. In catalytic methanol decomposition the elementary pathways were identified... 

Equation (4-5) can be directly utilized in statistical mechanical Monte Carlo and molecular dynamics simulations by choosing an appropriate QM model, balancing computational efficiency and accuracy, and MM force fields for biomacromolecules and the solvent water. Our group has extensively explored various QM/MM methods using different quantum models, ranging from semiempirical methods to ab initio molecular orbital and valence bond theories to density functional theory, applied to a wide range of applications in chemistry and biology. Some of these studies have been discussed before and they are not emphasized in this article. We focus on developments that have not been often discussed. 

Quantum mechanical methods follow a similar path, except that the starting point is the solution of the Schrodinger equation for the system under investigation. The most successful and widely used method is that of Density Functional Theory. Once again, a key point is the development of a realistic model that can serve as the input to the computer investigation. Energy minimization, molecular dynamics, and Monte Carlo methods can all be employed in this process. 

Ab initio molecular orbital methodology or density functional theory [158-160] would be suited for this combined QM/MM approach. However, in order to be able to compute the QM energies along the Monte Carlo simulation, nowadays a semiempirical Hamiltonian, like AMI [161], is a much more computationally efficient method. Before using AMI, the goodness of the semiempirical results in gas phase in comparison with the ab initio ones has to be tested. For systems in which the semiempirical results are poor, the relation... 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

During the past few decades, various theoretical models have been developed to explain the physical properties and to find key parameters for the prediction of the system behaviors. Recent technological trends focus toward integration of subsystem models in various scales, which entails examining the nanophysical properties, subsystem size, and scale-specified numerical analysis methods on system level performance. Multi-scale modeling components including quantum mechanical (i.e., density functional theory (DFT) and ab initio simulation), atom-istic/molecular (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational... 

The cornerstone of the field (the "Hartree-Fock" of Density Functional Theory) is the Local Density Approximation (LDA) also called the Local (Spin) Density (LSD) method Here the basic information on electron correlation, how electrons avoid each other, is taken from the uniform density electron gas Essentially exact calculations exist for this system (the Quantum Monte Carlo work of Ceperley and Alder) and this information from the homogeneous model is folded into the inhomogeneous case through the energy integral ... 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

Kinetic Monte Carlo and hyperdynamics methods have yet to be applied to processes involved in thermal barrier coating failure or even simpler model metal-ceramic or ceramic-ceramic interface degradation as a function of time. A hindrance to their application is lack of a clear consensus on how to describe the interatomic interactions by an analytic potential function. If instead, for lack of an analytic potential, one must resort to full-blown density functional theory to calculate the interatomic forces, this will become the bottleneck that will limit the size and complexity of systems one may examine, even with multiscale methods. 

We treat, in this chapter, mainly solid composed of water molecules such as ices and clathrate hydrates, and show recent significant contribution of simulation studies to our understanding of thermodynamic stability of those crystals in conjunction with structural morphology. Simulation technique adopted here is not limited to molecular dynamics (MD) and Monte Carlo (MC) simulations[l] but does include other method such as lattice dynamics. Electronic state as well as nucleus motion can be solved by the density functional theory[2]. Here we focus, however, our attention on the ambient condition where electronic state and character of the chemical bonds of individual molecules remain intact. Thus, we restrict ourselves to the usual simulation with intermolecular interactions given a priori. 

Essential progress has been made recently in the area of molecular level modeling of capillary condensation. The methods of grand canonical Monte Carlo (GCMC) simulations [4], molecular dynamics (MD) [5], and density functional theory (DFT) [6] are capable of generating hysteresis loops for sorption of simple fluids in model pores. In our previous publications (see [7] and references therein), we have shown that the non-local density functional theory (NLDFT) with properly chosen parameters of fluid-fluid and fluid-solid intermolecular interactions quantitatively predicts desorption branches of hysteretic isotherms of nitrogen and argon on reference MCM-41 samples with pore channels narrower than 5 nm. 

Modeling physical adsorption in confined spaces by Monte Carlo simulation or non-local density functional theory (DFT) has enjoyed increasing popularity as the basis for methods of characterizing porous solids. These methods proceed by first modeling the adsorption behavior of a gas/solid system for a distributed parameter, which may be pore size or adsorptive potential. These models are then used to determine the parameter distribution of a sample by inversion of the integral equation of adsorption, Eq. (1). 

However, it should be noted, that methods like the BJH, which are based on the macroscopic Kelvin equation, provide not a reliable basis for the calculation of pore widths < 5 nm [11,16,17]. Compared with new methods that rely on microscopic descriptions like the density functional theory (DFT) and Monte Carlo computer simulation (MC), the macroscopic thermodynamic methods underestimate the calculated pore diameter by ca. 1 nm [11,17]. In addition, it is argued that in case of nitrogen as adsorptive the BJH and related methods based on the Kelvin equation cannot be applied below a relative pressure p/po = 0.42 because this relative pressure is considered as the limit of the thermodynamic stability of liquid nitrogen. Hence, pore sizes obtained by application of the BJH method should be regarded as apparent rather than real pore sizes [7]. 

It is by now clear that several mechanisms and phases are present in the adsorption process in micropores, due to the interplay between gas-gas and gas-solid interactions, depending on their geometry and size. For this reason all those methods assuming a particular pore-filling mechanism, or adsorption model, should show shortcomings in some regions of the relevant parameters and their predictions should be compared with those based on more fundamental formulations of the adsorption process, like Density Functional Theory (DFT) [13] or Monte Carlo simulation [13,15]. Then, one question that arises is how the adsorption model affects the determination of the MSD ... 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

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