48 lines
4.7 KiB
TeX
48 lines
4.7 KiB
TeX
By solving the Schr\"odinger equation, one can predict some of the chemistry and most of the physics of a given system.
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However, although this statement is true and philosophically important, it was realised many years ago that, for more than \textit{one} electron, it is usually far too difficult from the mathematical point of view to solve this mighty equation.
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As Dirac pointed out,
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\begin{quote}
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\textit{``The aim of science is to make difficult things understandable in a simpler way.''}
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\end{quote}
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Consequently, it is essential to develop simple approximations that are accurate enough to have chemical and physical usefulness.
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To do this, quantum chemists and physicists have developed a variety of simple models that, despite their simplicity, contain the key physics of more complicated and realistic systems.
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A few examples are:
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\begin{itemize}
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\item the Born-Oppenheimer model: the motions of nuclei and electrons are independent;
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\item the orbital model: electrons occupy orbitals and move independently of one another;
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\item the local density model: the molecular electron density is built as an assembly of uniform electron gas densities.
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\end{itemize}
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Nowadays, all of these models are routinely applied in theoretical and/or computational studies.
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Spherical models are another example.
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One of the most popular starting points for modelling complex real life phenomenon by a highly simplified scientific model is the spherical geometry, and the most famous illustration of this is probably the so-called \textit{spherical cow} (Fig.~\ref{fig:cow}).
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While appearing completely nonsensical to most people outside the scientific area, these spherical models can be extremely powerful for understanding, explaining and even predicting physical and chemical phenomena in a wide range of disciplines of physics and chemistry.
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Besides, they offer unparalleled mathematical simplicity, while retaining much of the key physics.
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An explicit example is the spherical model introduced by Haldane \cite{Haldane83} to explain the fractional quantum Hall effect (FQHE), for which Laughlin, St\"ormer and Tsui received the Nobel prize in physics.
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This geometry has been instrumental in establishing the validity of the FQHE theory, and provides the cleanest proof for many properties.
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In this chapter, we will show that the spherical geometry can be also useful to better understand the structure of the exact electronic wave function.
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Almost ten years ago, following this idea, we undertook a comprehensive study of two electrons on the surface of a sphere of radius $R$ \cite{TEOAS09, Concentric10, Hook10}.
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We used quantum chemistry electronic structure models ranging from HF to state-of-the-art explicitly correlated treatments, the last of which leads to near-exact wave functions and energies.
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This helped us to understand not only the complicated relative motion of electrons, but also the errors inherent to each method.
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It eventually led to the important discovery that the system composed of two electrons restricted to the surface of a $D$-sphere (where $D$ is the dimensionality of the surface of the sphere) is exactly solvable for a countable infinite set of values of $R$ \cite{QuasiExact09, ExSpherium10, QR12, ConcentricExact14}.
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In other words, it means that the \textit{exact} solution of the Schr\"odinger equation can be obtained for certain ``magic'' values of the radius of the sphere \cite{QuasiExact09}.
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This discovery propelled the two-electrons-on-a-sphere model (subsequently named spherium), into the exclusive family of exactly solvable two-electron models.
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Moreover, after an exhaustive study of the ground state of two electrons confined by various external potentials \cite{EcLimit09, Ballium10}, we noticed that the correlation energy is weakly dependent on the external potential, and we conjectured that the behaviour of the two-electron correlation energy, in the limit of large dimension, is \textit{universal}!
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The rigorous proof of this conjecture has been published in Ref.~\cite{EcProof10}.
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In particular, we showed that the limiting correlation energy at high-density in helium and spherium are amazingly similar \cite{Frontiers10}.
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However, while the closed-form expression of the limiting correlation energy has never been found for helium, the value for spherium is quite simple to obtain.
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This shows the superiority of the spherical geometry approach and that it can be used in quantum chemistry to provide robust and trustworthy models for understanding, studying and explaining ``real world'' chemical systems.
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In this chapter, we will summarise some of our key discoveries.
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%%% FIG 1 %%%
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\begin{figure}
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\centering
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\includegraphics[width=0.2\textwidth]{../Chapter2/fig/sphericalcow}
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\caption{\label{fig:cow} A spherical cow.}
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\end{figure}
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%%% %%%
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