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75

What are the state-of-the-art algorithms for long-integer multiplication? First let me address the point you raised about the schoolbook algorithm having $\mathcal{O}(n^2)$ scaling, by saying that this was not the state-of-the-art algorithm used in most matter modeling software. Below I give a brief overview: (1960) Karatsuba multiplication. $\mathcal{O}(n^{...


32

Kohn is easily one of my favorite humans of all time, and he was a role model to whom I looked up in great admiration for most of my academic life; in fact before this site was created, I proposed that we name it after him. However I completely disagree with the sentence that you have quoted. Keep in mind that even though the Nobel Lecture was in 1999, Kohn ...


31

This $O(n\ln n)$ integer multiplication algorithm is a galactic algorithm, meaning that it won't be used despite being "of lower complexity" because it only becomes more efficient than existing algorithms for problems vastly larger than any relevant to us in practice. The problem is big-$O$ notation only tells us how the algorithm behaves for ...


27

2006 (Grimme): Double hybrid functionals The timeline of milestones you have given, includes a hybrid functional called B3LYP, which mixes a Hartree-Fock exchange functional with a GGA exchange-functional. In 2006, Stefan Grimme introduced what later became known as "double hybrid functionals", which not only mix the Hartree-Fock exchange functional with a ...


25

2015 (Sun et al.): SCAN functional The SCAN meta-GGA functional is an extension of the popular PBE GGA [1] and the TPSS [2] and revTPSS [3] meta-GGAs, SCAN adheres to all 17 known exact XC constraints and is constructed to be almost exact for the noble gasses and jellium surfaces. Early evidence suggests that SCAN is more accurate than and of comparable ...


21

To take a slight detour, we can also look at the progress of matrix multiplication algorithms. As mentioned in a few comments here, standard matrix multiplication is $O(n^{3})$ and any exact method for a general matrix is going to require $O(n^{2})$ operations just to process all the elements of initial matrices. Over the last 50 years, different methods ...


20

2013: Density-Corrected DFT (DC-DFT) The goal of Density-Corrected DFT (DC-DFT) is not only to get better accuracy but also to understand and correct the true error in the functional approximation.[1,2] In any approximate density functional, the DFT error is $\Delta E = \tilde E[\tilde n] - E[n]$ where $E$ and $n$ are exact functional and density while $\...


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2004 (Yanai et al.): Range separation Often, the source of DFT improvement comes from Hartree-Fock as is also obvious from the answer involving double hybrid functionals. So too it is with range-separation. The electron-electron Coulomb operator for the exchange contribution is separated into a short and long range contribution. \begin{equation} \frac{1}{...


19

"Matter modeling" is a term that was coined by Stack Exchangers! That's why the words "matter" and "modeling" next to each other, don't seem to exist in any of the results in a Google search of "Matter Modeling", apart from this Stack Exchange site, Adam Iaizzi's blog post about this SE site, and maybe our community's ...


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1993 (Becke): Hybrid Functionals Axel D. Becke introduced the adiabatic-connection model, which allows for mixing of DFT exchange and Fock-like exchange via the formula $$ E_{\text{x}} = a \cdot E^{\text{HF}}_x + b \cdot E^{\text{GGA}}_x $$ to obtain the exchange part of the exchange-correlation energy. Typically, one imposes $a+b = 1$, but some authors ...


17

1995 (Casida): TD-DFRT Time-Dependent Density Functional Response Theory is a linear response formulation of TDDFT for the calculation of excitation energies and corresponding transition amplitudes, that in turn allows to evaluate electronic spectra of molecular and condensed matter systems. The time-dependent density functional theory (TDDFT) in the Kohn–...


17

Can someone explain in detail the impact of any of the multiplication algorithms scaling better than N2, for some practical application? An actual application is right in front of our eyes: digital signature using RSA. If I click on the lock icon for the present page in my browser, then on the arrow on the right of Connection secure, then More Information, ...


15

Even the simplest better-than-schoolbook (O(n^2)) algorithms like Karatsuba are only useful in practice for large n. But what is n? It's not single bits, and it's not decimal digits. (Posting this tangent as requested in comments.) Software implementations of an extended-precision multiply algorithm work in integer chunks as wide as the hardware provides. ...


13

Practical importance: compactifying explanations It is widely believed that $\mathcal{O}(n \log n)$ is the best possible result, and therefore we no longer have to say $\mathcal{O}(n\log n\cdot 2^{2\log^*n})$ every single time in every single paper in related fields, we can just say $\mathcal{O}(n \log n)$ every time now. Here is a related quote from Reddit:...


13

2005 (Bartlett): ab initio DFT Basically one takes the xc functional from a wave function approach such as MBPT(2), CC, etc. and constructs an xc potential from them using density conditions or a functional derivatives approach. Summary of developments is best captured in the following article: "Adventures in DFT by a wave function theorist." Details on ...


12

Constraint #13: Size-Extensivity While the Wikipedia page for size-consistency and size-extensivity gives a clear formula for the definition of size-consistency, unfortunately they did not give a definition of size-extensivity, so I had to look deeper into the reference that they provided. They say that size-extensivity was introduced by Bartlett, and they ...


11

I would add some developments in TDDFT that came around 1996 and resonated only later such as: the Casida equation (Casida 1995) that allows to calculate excitation energies and electronic spectra the real-time TDDFT (Yabana & Bertsch 1996) a non-perturbative TDDFT technique in which the time-dependent Kohn–Sham equation is solved by direct propagation ...


10

First of all, we assume the matter is constituted by nuclei and electrons, as illustrated by the following figure: Mathematically, the matter is mapped into the following Hamiltonian: $$H=T_e + T_n + V_{ee} + V_{nn} + V_{en} \tag{1}$$ which is called the standard model of condensed matter physics. In principle, all information (ground state/excited state/...


9

Most chemists' point of view could be condensed as follows: Implementation of DFT in Gaussian (Pople et al, 1992) LDAs and GGAs were implemented in Gaussian 92/DFT by Pople, Gill and Johnson [Chem Phys Lett 199, 557 (1992)]. DFT better than ab initio (Johnson et al, 1993) BLYP was found to yield more accurate equilibrium geometries, dipole moments, ...


9

In terms of the role of chemists for what you call the "most essential task": "What I understand is that the most essential task is the improvement of exchange correlation functionals to give better and more accurate results" "... the development of them is a job for a quantum physicist for example." I am reminded of two ...


8

As a computational chemist, I find my role is to act as a translator. In a molecular dynamics simulation, I need to know enough physics and maths to translate chemical structures into a physical model, and make sure that the physics at play produces a model that makes sense chemically. Then if I'm doing multiscale modelling, say at finite element level, I ...


8

The Last Question by Isaac Asimov is a short story where a computer is tasked with solving the problem of entropy. It's a brilliant story and definitely worth listening to and reading https://youtu.be/ojEq-tTjcc0


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I don't think analog computers are quite at the level they need to be yet, but there are people working to make them applicable to matter modeling applications. Dr. Rahul Sarpeshkar at MIT developed simple circuits to model a particular pair of cellular protein production processes and mimic the differing pathways to the same result. They cite the continuous ...


7

The role of the chemist in computational chemistry is to ask the right questions. Then, the computational chemist can step in, and start computing. The computational chemist should know which models are good for the problem in question, and what kinds of pitfalls to watch out for. Then, at the very deep end you have the method developers as well as the code ...


7

2020 Furness et al: r$^2$SCAN functional The SCAN functional is the most recent meta-GGA functional constructed from first principles, which satisfies all known bounds. However, SCAN is also numerically pathological: getting converged energies requires huge quadrature grids, and constructing pseudopotentials is difficult. This motivated the construction of ...


7

1997 (Marzari & Vanderbilt): MLWF These methods enable a more qualitative view of the electron density by projecting the Bloch wavefunctions into localized Wannier functions [1], which is especially useful when it comes to transition metal systems, but not limited to these. This description enables DFT practitioners to "talk" to the modelling ...


7

Disclaimer: These are just some observations; I don't claim to know the answer to this question. There's a lot of overlap between both camps, but I think people tend to stick mostly with one kind of system, in their research. You may be largely correct - I'd just like to point out that if we are looking at the simulation engines (rather than what people do)...


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2001 (Taylor et al.): DFT+NEGF Combining density functional theory (NEGF) with nonequilibrium Green's function method (NEGF), a self-consistent first-principles technique for modeling quantum transport properties of atomic and molecular scale nanoelectronic devices under external bias potentials is reported. Implementation packages: QuantumATK, Nanodcal ...


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2014 (Gagliardi): MPCDFT Multiconfiguration Pair-Density Functional Theory (MC-PDFT) is a theoretical framework that combines multiconfigurational wave functions with a generalization of density functional theory. As the reference wavefunction is multiconfigurational rather than being a single Slater determinant, it has the advantage that it can describe ...


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