19

This is an excellent question! Reversibility in MD is useful because: Time-reversibility in a numerical integrator leads to a doubling of the accuracy order (see Propositions 5.2 and Theorem 6.2 here). Reversible maps can be readily Metropolized, for example in a hybrid Monte-Carlo scheme. This gives an easy way to enhance the sampling efficiency and ...


15

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 ...


15

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 ...


11

I would argue the main reason this is important is philosophical, linked to the history of science and determinism (as proposed by Laplace). Newtonian mechanics is mathematically reversable while any observation in a "real world" system is one of irreversibility and increasing entropy, which is why we end up with Loschmidt paradox. From a ...


10

Seeing that this question has gathered attention but no replies, I will give it a stab. Note that I am not an expert on DFT or functional calculus, so take this with a grain of salt. As usual, suggestions to the post will be welcome! Using an approach I saw here, we can use a chain rule and obtain the following: $$\frac{\delta F[\rho(\boldsymbol{r})]}{\delta ...


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

You are correct that this is due to not including quantum effects. Ref 1 in your figure is the paper cited below. In this paper, they explicitly mention that $C_v$ calculated using the cell-cluster method is in good agreement only for sufficiently high reduced-temperatures. From section IV of this paper: ...the calculation is in acceptable agreement with ...


8

Short introduction to ergodicity Ergodicity is when the time-average equals the ensemble-average. A process is ergodic if the time-average "converges in the square mean" to the ensemble average. A sequence $X_t$ converges in the square mean to $X$ if: $$ \tag{1} \lim_{t\rightarrow \infty}\langle \left|X_t - X\right|^2 \rangle = 0, $$ where $\...


8

It's very hard to answer this definitively unless you are a bit more specific. However, to speak somewhat generally, it is the case that NVE and NVT ensembles become equivalent in certain thermodynamic limits (namely an infinite number of particles). In practice, however, one chooses the ensemble based on the free energy you are interested in sampling, or ...


7

As with all MD simulations, you have to assume (often wrongly) convergence with finite time. This is fairly easy to do with autocorrelation functions though, because you know that once they become negative, you are already in the random fluctuation (i.e. uncorrelated noise) zone and this is the point where you can stop. This is slightly trickier in some ...


7

It is straightforward to show that in a typical $NPT$ setting the Zwanzig equation still only depends on the energy difference and not on the volume (here I define $H$ to be the Hamiltonian of each system, respectively and $x$ to represent all variables over phase space): $$\frac{Z_{B,NPT}}{Z_{A,NPT}} = \int \int e^{-\beta \Delta H(x)} \frac{e^{-\beta H_A(x)...


7

Thermochemistry (for a single molecule, e.g., ideal gas) depends on the Temperature. At T=0K thermochemistry is kind to us. At T = 0K $U = G = H$. This is because H = U + PV, which for an ideal gas is H = U + RT, and T = 0K, and also, G = H-TS, and again, T = 0K so that cancels The only direct calculation a QM program is doing, that I am aware of, is ...


6

Energy in an NPT simulation is not conserved, but (once equilibrated), it will fluctuate around an average value, and that average value has meaning. That is the ensemble average for your NPT and is a valuable and useful property. You are also correct that the internal energy is the summation of potential and kinetic contributions. To be thorough, pp. 60 of ...


5

It definitely can matter. The best thing to do is to consider the experiment you are comparing to, even if that experiment hasn't actually been done yet. In many cases, that means you want to run dynamics in the NPT ensemble (since bench experiments tend to be at fixed pressure and temperature). Here's an example where we know we won't get the same results ...


5

Ensembles are essentially artificial constructs. In the thermodynamic limit (for an infinite system size) and as long as we avoid the neighborhood of phase transitions it is generally believed that there is an equivalence between ensembles. A consequence of the equivalence of ensembles is that the basic thermodynamic properties of a model system may be ...


5

I know it's a boring thing to say, but: It depends on what you want to do. The way I use Langevin thermostats, is to ensure good equipartitioning in my setup, so that I don't have any local hotspots in the system or something like that. If heat transfer is slow in the system, "global" thermostats like Nosé-Hoover and the like will equilibrate ...


5

There are two possible things tripping you up here: Phonons are collective oscillations: they involve the motion of all the atoms together. Therefore it only makes sense to talk about the phonons of the whole system, not any individual atom. The density of states only makes sense as you take $N\to \infty$. For finite $N$, there is a finite/discrete number ...


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 ...


4

"This works for organic molecules, but what happens when the excited states are closer in energy to the ground state, for example in open-shell molecules or in atoms?" If there's excited states close to the ground state, the approximation you said Gaussian uses, where excited state contributions are neglected, seems not to be such a great idea ...


4

Before working through the equations, I'll try to explain the logic behind what they are doing. It helps to think of think of their process backwards and assume they want an expression like equation 26: $$\mathbf{F}_I(\mathbf{R}^N)=\langle\mathcal{F}_I(\mathbf{r}^n)\rangle_{\mathbf{R}^N} \tag{7}\label{7}$$ where the left-hand side is the coarse-grained ...


3

The paper bellow can give you some insides. Using a Computer Tomography recording, they build a 3D model for computer simulation using FEM (Finite Element Method) and then do a 3D printing. The article: T. Vampola, J. Horáček, V. Radolf, J. G. Švec, and A. Laukkanen, Influence of nasal cavities on voice quality: Computer simulations and experiments, J. ...


3

So what happens when the contribution of excited determinants are included in the vibrational and rotational terms? Nothing, except that the Hessian and/or gradient are more difficult to evaluate, for example see coupled-perturbed Hartree-Fock. I think maybe your confusion is arising because the excited determinants are not exactly excited-states. A multi-...


3

Aren't you missing the volume in the denominator? And the order of the norm and average is probably off in the second term. The original equation should be $$ \epsilon = 1 + \frac{\langle |\mathbf{M}|^2\rangle - |\langle\mathbf{M}\rangle|^2} {3\epsilon_0 V k_B T} $$ and in units where coulombs constant is one, $k = 1 = (4\pi\epsilon_0)^{-1}$, you get $1/\...


3

Molecules will not be on both sides of the box at once because this is explicitly prevented by most good MD packages. You can calculate distances which take into consideration the PBC. For example, here is a code to calculate all the pairwise distances with periodic boundary conditions (x_size = [16,16,16]) This is modified from periodic boundary conditions ...


3

Translation diffusion coefficient $D_{t}$ may be calculated by linear fitting of the MSD of center of mass of a molecule: $$MSD(\tau)=<(\textbf{r}(t+\tau)-\textbf{r}(t))^{2}>$$ $$D_{t}=\frac{MSD(\tau)}{6\tau}$$ where $\tau$ is the lag time between the two positions. One should also be aware of the artifacts from the periodic boundary conditions (is ...


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