Expanding the scope and scale of first-principles quantum-mechanical simulations with the ONETEP linear-scaling method on high performance computers

计算机模拟在我们的社会中扮演着重要的角色，例如，飞行模拟器可以让飞行员接受比在空中更便宜、更安全的训练。在科学技术领域，计算机模拟是理解和预测材料复杂过程的有力工具。模拟通常与传统实验一起使用，但当实验过于昂贵或甚至无法进行时，也可以使用模拟，例如，当研究极端条件下的材料时，例如地球中心的高温高压。随着量子力学（QM）的发现，上个世纪之交见证了科学革命的开始。在非常小的尺度上，大自然的行为方式与我们的日常体验截然不同。如果我们缩小到一个原子的大小，导航将变得非常困难，因为不确定性原理说，不可能在同一时刻准确地知道你在哪里和你要去哪里！尽管有这种奇怪的行为，量子力学还是惊人地准确，并为我们所有的科学和技术提供了基础。除非量子力学方程能够解决当今科学家和工程师感兴趣的问题，否则这样的主张是没有用的。挑战在于方程式非常复杂——即使是两个电子也很难在纸上找到答案。在某种程度上，量子力学本身提供了答案，因为它导致了晶体管的发明，从而导致了计算机的发明。然而，即使在最快的计算机上，也只能精确地解决小分子的量子力学方程，而今天科学家感兴趣的系统涉及数千个。由于问题的规模，即使计算机技术的快速和不懈的进步也不能提供全部的答案。完成某项任务所需的工作量通常会随着它的大小而增加，例如，修剪草坪所花费的时间与它的面积成正比：如果你把花园的面积扩大一倍，你将花费两倍的时间。这是线性扩展的一个例子，但是许多任务所涉及的工作量比这增长得更快。按字母顺序整理一套书或cd，或在纸牌游戏中安排你的手通常是所涉及物体数量的平方：如果你把数字翻三倍，将需要9倍（3的平方）的时间。还有一些任务要糟糕得多，比如解决旅行推销员问题，找到到达给定地点的最快路线。再增加一个地点，解决问题的时间就会增加一倍。即使你能在一分钟内解决三个地点的问题，22个地点也需要你一整年的时间。求解QM的方程就像这样。然而，在20世纪60年代，密度泛函理论（DFT）的引入取得了重大飞跃，沃尔特·科恩（Walter Kohn）因此获得了1998年诺贝尔化学奖。不利的标度的起源是电子是带电粒子。同种电荷相斥，所以一个电子的轨迹取决于所有其他电子的轨迹，因为它想要避开它们。所以当涉及到更多的电子时，求解描述这些轨迹的方程变得更加困难。但是DFT的显著结果是整个系统的物理性质，我们想要的问题的答案，不依赖于这些单个轨迹的细节，而只依赖于平均值。因此，方程的线性缩放解是可能的，这有望极大地扩大量子模拟的规模。这项工作的目的是改编最近开发的线性缩放DFT代码，称为ONETEP，以利用计算机技术的最新发展，使其在目前最强大的计算机上有效运行。利用线性缩放方法和现代计算机的力量，科学家们将首次对由数万个原子组成的系统进行基于量子力学的模拟。

Computer simulations play an important part in our society e.g. flight simulators allow pilots to be trained more cheaply and safely than in the air. In science and technology, computer simulation is a powerful tool for understanding and predicting complex processes in materials. Simulations are often used alongside conventional experiments, but they can also be used when experiments are too expensive or even impossible to perform, e.g. when studying materials in extreme conditions such as the high temperatures and pressures at the centre of the Earth.The turn of the last century saw the start of a scientific revolution with the discovery of quantum mechanics (QM). On very small scales, nature behaves in a radically different way from our everyday experience. If we were shrunk down to the size of an atom navigation would become very difficult, as the uncertainty principle says that it is impossible to know at the same moment precisely where you are and where you are going! In spite of this bizarre behaviour, QM is astonishingly accurate, and provides the foundation for all our science and technology.Such claims are of no use unless the equations of QM can be solved for problems of interest to scientists and engineers today. The challenge is that the equations are very complicated - even two electrons are too much for finding a solution on paper. In a way, QM has itself provided the answer as it led to the invention of the transistor and so to the computer. However, even on the fastest computers it is only possible to solve the equations of QM exactly for small molecules, whereas the systems of interest to scientists today involve many thousands. Even the rapid and relentless progress of computer technology cannot provide the whole answer, because of the scaling of the problem.The work needed to accomplish a certain task generally increases with its size e.g. the time taken to mow a lawn is proportional to its area: if you double the size of your garden it will take you twice as long. This is an example of linear scaling, but the effort involved in many tasks increases faster than this. Sorting a set of books or CDs into alphabetical order or arranging your hand in a game of cards usually scales as the square of the number of objects involved: if you triple the number it will take nine (three squared) times as long. There are some tasks which are much worse, such as solving the travelling salesman problem to find the quickest route to visit a given set of places. Adding one more location doubles the time it takes to solve. Even if you can solve the problem for three locations in one minute, just 22 will take you a whole year. Solving the equations of QM exactly scales like this. However, in the 1960s a significant leap forwards was made with the introduction of density-functional theory (DFT), for which Walter Kohn won the 1998 Nobel Prize in chemistry. The origin of the unfavourable scaling is that electrons are charged particles. Like charges repel, so one electron's trajectory depends on all the others', as it wants to avoid them. So solving the equations which describe these trajectories becomes much harder as more electrons are involved. But the remarkable result of DFT is that the physical properties of the whole system, the answers to the questions we want, do not depend upon the details of these individual trajectories, but only on the average. So a linear-scaling solution of the equations is possible, which promises dramatically to expand the scale of quantum simulations accessible.The aim of this work is to adapt a recently-developed linear-scaling DFT code called ONETEP to take advantage of recent developments in computer technology so that it runs efficiently on the most powerful computers now available. Harnessing the power of linear-scaling methods and modern computers will allow scientists to perform simulations based on QM for systems made of tens of thousands of atoms for the first time.