# Tag Info

17

Another extremely popular resource is Frenkel and Smit's textbook "Understanding Molecular Simulation". It covers all basics on molecular dynamics, Monte Carlo, some common enhanced sampling methods, free energies and it even derives the Ewald summation. This is basically a book that doesn't shy away from messy derivations and it gives you a lot of ...

16

A good place to start is the classical Allen-Tildesley book, Computer Simulation of Liquids, which covers the basics of molecular dynamics that hasn't really changed in a long time. The book can be supplemented with literature, such as review articles for whatever it is you want to do. Software manuals are also often quite useful to find out how things are ...

13

I will outline the way it was derived in the original 1990 paper. We start with an ansatz for the time-dependent wavefunction: $$\tag{1} \psi(x_1,\ldots,x_n;t) = \sum_{j_1=1}^{m_1}\cdots \sum_{j_n=1}^{m_n}a_{j_1\cdots j_n}\phi_{j_1}^{(1)}(x_1,t)\cdots \phi_{j_n}^{(n)}(x_n,t),$$ with single-particle functions (SPFs) satisfying (...

10

Another good book that starts from the very beginning and it's very hands-on is "The art of molecular dynamics" by D. C. Rapaport. It is particularly useful if you want to code up a MD code yourself. This is something I always recommend, even if you want to go on and use one of the big MD packages out there, since you will learn the essentials and ...

9

I'll try to give a short but reasonably rigorous way of thinking about the exactness of density functional theory (DFT). Consider $N$ electrons under the influence of a fixed external potential $v(\mathbf{r})$ for which the ground state electron density is $n(\mathbf{r})$. The external potential might be a sum of individual potentials from atomic nuclei, but ...

7

The part of the paper where they give the formula for the populations is here: "The timedependent (TD) vibronic wavefunction of the system is computed solving the TD Schroedinger Equation. It is written as $|\Psi_i(\mathbf{q},t)\rangle = \sum_i |d_i\rangle |\Psi(\mathbf{q},t)\rangle$ so that electronic populations are simply $P_i(t) = \langle \Psi_i (... 7 The Monte Carlo Sign Problem For classical systems, Monte Carlo works extremely well. Quantum Monte Carlo is very powerful, but there are many interesting systems that suffer from the sign problem, which makes Monte Carlo exponentially hard. Rather than discuss this in detail here, see the discussion on the sign problem on the physics SE. 5 I believe that Eq.13 of the same reference is the construction you are asking about. Since MCTDH is an initial value problem, you have to construct the SPFs at the initial time by choosing the initial value of the time-dependent coefficients$a^{(\kappa)}_{kj}$and then MCTDH will propagate these coefficients. Then your question becomes "how to ... 4 This response comes late, but I hope you or other readers will find it useful. Regarding your question on how to derive MCTDH equations (directly) from the time-dependent Schrödinger equation (SE), I would add the following comments. The SE is exact in the full Hilbert space, but for numerical simulations, you need to choose a finite (usually small) working ... 4 Generally speaking, Ehrenfest dynamics tries to model nonadiabatic processes. These are processes where an electronic wavepacket propagates on several potential energy surfaces. The different surfaces have different forces associated to them. This means that the parts of the wavepacket on the different surfaces move in different directions. Ultimately, this ... 4 RPMD (Ring Polymer Molecular Dynamics) Introduction (correlation functions and Kubo transforms): For a time-dependent quantum operator$\hat{A}(t)$the auto-correlation function of the operator at time$t$versus at time 0 can be written as:$\$ c_{AA}(t) \equiv \frac{\textrm{tr}\left( e^{-\beta H}\hat{A}(0)\hat{A}(t) \right)}{\textrm{tr}\left({e^{-\beta H}}\...

4

Also worth mentioning Bill Hoover's excellent textbooks, Molecular Dynamics which provide an introduction and a slightly different perspective on molecular dynamics, along with Computational Statistical Mechanics. They are freely downloadable from his website. I also second Tuckerman's "Statistical Mechanics: Theory and Molecular Simulation" as one ...

2

Since you are interested in quantum simulation, I would actually suggest reading Feynman and Hibbs Quantum Mechanics and Path Integrals. It is very cheap which is a big plus and also remarkably readable as everything Feynman wrote is. Even though the book would be considered advanced because path integrals are considered advanced, much of what the book does ...

1

I don't know of any exact methods, but quadpy is a really good place to start, and possibly ask the related question there. Not really an answer, but more of a hint.

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