Computational Physics Thijssen Pdf 32 Fixed

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The code is the best documentation. If interested only in the computational physics part, you have to look only in one namespace: Scattering, also perhaps in the SpecialFunctions namespace, but the last version uses the std implementations. The class names are very suggestive, LennardJonesPotential, Numerov and Scattering, to name some. I hope the code expresses intent well enough to not need much description.

The fundamental relevance and concrete applications of the confined quantum systems have been established since the early days of quantum physics [13, 28, 41], up to the present day [23, 35, 39, 40]. Simple confined physical systems have a very long tradition in physics because of their importance as basic theoretical problems [4, 7, 24,25,26,27], and as prototypes to gain insights in semiconductor physics (quantum dots [8, 33], quantum wires and quantum wells [17, 30]). Atoms imprisoned in zeolite traps, clusters and fullerene cages provide good examples of confined systems in atomic physics and inorganic chemistry [6, 10, 18, 42].

represents a diffusion process. To implement Eq. (2), we use a special technique: the central limit theorem of probability theory guarantees that, in the repeated application of random displacements, a Gaussian distribution like \(P_{1}\) is obtained anyway. Consequently, our programs do not use computationally expensive Gaussian random generators, but shift each coordinate according to a uniform probability distribution with the same variance as \(P_{1}\):

In Fig. 2, the comparison with the best available results, in the case of the unconfined system, shows how a very simple DMC code algorithm is able to obtain a satisfactory prediction with small deviations from the reference points in most cases. By inspecting the figure, it is also clear that the trend of the calculated solution is satisfactory: therefore, in a future application to ab initio molecular dynamics, the method may be exploited to produce a reliable interpolated potential energy surface. The potential energy curve is actually reproduced with sufficient accuracy to perform, for example, a posteriori calculations of vibrational states. This is not a calculation that can be done with our computational code (furthermore, it is out of the domain of the DMC method studied here), because the free collinear geometry is not stable in a vibration (the geometry of the free \(\hbox {H}_{3}^{+}\) ion is \(D_{3h}).\)

As a preliminary check, we have run the code in the case of the ground state of \(\hbox {H}_{2}\). At the equilibrium internuclear distance of 1.40 a.u., with a \(10^{3}\) walkers and 50,000 times steps at \(\tau = 0.002\), our code provides a ground energy of \(-1.171\) a.u., to be compared with the exact value given by \(-1.1744\) a.u. The accuracy can be further improved with a higher computational cost.

In conclusion, in the context of a new emerging class of problems, our proposal of the unbiased diffusion Monte Carlo calculation technique is based on the contrast between analytical fatigue and computational cost. The variance reduction techniques used for Monte Carlo methods require the formulation of fully correlated tentative wavefunctions: this formulation is not immediate in the case of confined systems, and it is not immediately transportable from one confinement geometry to another. On the other hand, the accuracy of the unbiased Monte Carlo method can be increased by accurate statistical analysis and simple solutions. Nowadays, the large availability of fast computing resources, even portable ones, might lead a researcher to favor an approach that allows to develop native, short and readable programs and, at the same time, to explore different confinement geometries with minimal modifications. Even with the limits due to statistical fluctuations, such algorithm takes the correlations between electrons into full account. Furthermore, our technique, despite being in the spirit of the primitive one, includes improvements of straightforward implementation, such as the faster propagator and the method of restarting the walker cloud. Ultimately, our experience suggests that a simpler technique, employing no analytic functions of significant complexity, can be a resource in the exploration of the wide range of possibilities that has been opened up by the recent reconsideration of confined quantum systems.

It isn't a book just on Monte Carlo methods or just about C++, but a good book about general computational physics that has a few chapters on Monte Carlo methods including quantum Monte Carlo is Computational Physics by J.M. Thijssen.

@article{PaoloDiSia2014,abstract = {This paper presents an overview of mathematical models for a better understanding of mechanical processes, as well as dynamics, at the nanoscale. After a short introduction related to semi-empirical and ab initio formulations, molecular dynamics simulations, atomic-scale finite element method, multiscale computational methods, the paper focuses on the Drude-Lorentz type models for the study of dynamics, considering the results of a recently appeared generalization of them for the nanoscale domain. The theoretical framework is illustrated and some examples are considered.},author = {Paolo Di Sia},journal = {Nanoscale Systems: Mathematical Modeling, Theory and Applications},keywords = {Theoretical Modeling; Drude-Lorentz type Models; Computational Methods; Finite Element Methods; Nano-Bio-Technology},language = {eng},number = {1},pages = {null},title = {Overview of Drude-Lorentz type models and their applications},url = { },volume = {3},year = {2014},}

TY - JOURAU - Paolo Di SiaTI - Overview of Drude-Lorentz type models and their applicationsJO - Nanoscale Systems: Mathematical Modeling, Theory and ApplicationsPY - 2014VL - 3IS - 1SP - nullAB - This paper presents an overview of mathematical models for a better understanding of mechanical processes, as well as dynamics, at the nanoscale. After a short introduction related to semi-empirical and ab initio formulations, molecular dynamics simulations, atomic-scale finite element method, multiscale computational methods, the paper focuses on the Drude-Lorentz type models for the study of dynamics, considering the results of a recently appeared generalization of them for the nanoscale domain. The theoretical framework is illustrated and some examples are considered.LA - engKW - Theoretical Modeling; Drude-Lorentz type Models; Computational Methods; Finite Element Methods; Nano-Bio-TechnologyUR - ER - 2b1af7f3a8