# Tag Info

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Disclaimer and warning: long and likely biased answer. Background: The Pople style basis sets were defined almost 50 years ago. The 6-31G was designed for HF calculations, the 6-311G for MP2 calculations. For computational efficiency reasons, the s- and p-exponents were constrained to be identical. Polarization functions were defined for 1d, 2d, 3d and 1f. ...

17

There are two considerations guiding the choice a basis for orbital expansion: 1. Compactness; 2. Efficiency of computations. There are two common choices for basis functions (A) Gaussians and (B) Plane waves. Both of those allow for the most efficient way to evaluate the integrals needed to construct the Fock matrix (i.e. second derivative for kinetic ...

16

I would like to expand on Roman Korol's answer a bit. He already lists GTO's and plane waves as they are the most common kind of basis functions. These are characteristic for the underlying models by which they are motivated. GTO's approximate the solutions to the hydrogen atom and are thus atom centered functions used for molecules. Plane waves on the other ...

15

The justification is simple and comes from a very fundamental law of thermodynamics: Internal energy is a complete differential form and is independent of intermediate states and only depends on start and final states: $$\Delta U = U_{2} - U_{1}$$ In your case: $U_{2} = E_{AB}$ and $U_{1} = E_{A} + E_{B}$ and if $\Delta U$ or binding energy $E_{\text{... 14 There's two facets to this question: What methods can be used for excited states in crystals? (the title, and final sentence) What methods can be used for excited states with plane waves? (paragraphs 1 and 3) I will begin by answering the second question, which is in some sense grounded by this assumption: "it is not possible, as far as I know, to do ... 13 Another slightly less commonly seen basis are the sinc functions, which are related to plane waves, but come at the problem from the perspective of a position, rather than momentum, space. They are delocalized functions, but sharply peaked at their center point and zero valued at the centers of other sinc functions, effectively partitioning space into a grid.... 13 The main positives: you can do all-electron calculations you don't need to set up pseudopotentials / PAWs you can study core properties you can use hybrid functionals cheaper / run post-HF calculations The negatives: basis set is geometry dependent, so you get superposition error it's harder to get results close to the complete basis set limit Either ... 12 In the example you highlighted and indeed in most plane-wave DFT codes, there is periodicity in all three dimensions including for surface slab calculations. In the case of a surface slab, vacuum space is commonly added in the$z$dimension. The vacuum space is there so that an adsorbate can bind of course, but it's also there because of the boundary ... 12 The clearest example in my mind is if you want to understand the orbital-based contributions to some phenomena (e.g. bonding, a reaction energy), particularly if the periodic material being modeled is more like a molecular solid where the chemical picture of orbitals is more intuitive than bands. There are several schemes out there that try to go from PAW to ... 12 Your supervisor is correct that in almost all practical cases in quantum chemistry, the cost savings of using Gaussian-type orbitals far outweigh the disadvantages of needing more orbitals. First of all, there is a bit of a misconception that Slater-type orbitals are far more accurate. The motivation for using Slater-type orbitals is due to their resemblance ... 12 Different people will have slightly different opinions, so take my answer with a grain of salt. I'll try to give the minimum publishable basis sets, i.e. if you use worse basis sets than these, you'll have a strong reason why you do so, written in the article, otherwise you'll probably face suspicion from the reviewers. DFT (excluding double hybrid) geometry ... 11 In my opinion a solid, brief overview of DFT is given here. https://www.archer.ac.uk/training/course-material/2014/04/PMMP_UCL/Slides/castep_1.pdf The pro/cons of plane waves and other basis sets are discussed and I will list them here in case the link goes dead. Pros: Fourier coefficients stored in regular grid. Efficient FFT algorithms between r- and ... 11 The answer in part is that not all the elements up through$\ce{Xe}$have (or had until fairly recently) a nonrelativistic correlation consistent basis set. Specifically, the alkali and alkaline earth elements (columns 1 and 2 of the periodic table) have been much slower to get nonrelativistic basis sets. The cc-pVnZ basis sets for$\ce{Li}, \text{ }\ce{Be}, ...

11

Pure plane-wave basis sets have the following advantages when used in periodic DFT (or HF) simulations: Orthogonal Computationally simple (operators with derivatives are particularly straightforward) Low-scaling methods allow easy transformations between real- and reciprocal-space Basis set size does not scale with electron count Independent of atomic ...

11

There are a lot of different methods, but a good starting point to understand how this might be done is the method of steepest descent. First we take the generalised eigenvalue expression you cite and rearrange to express the energy expectation value as $$E=\frac{\phi^\dagger \mathrm{H}\phi}{\phi^\dagger \mathrm{S}\phi},$$ where I've generalised to allow ...

11

First of all, MP2 (for example) is actually guaranteed to converge from above to the basis set limit even though MP2 in one specific basis set can give an energy lower than the FCI energy in that same basis set. So it's variational character in the basis set sense that matters here. Furthermore, variational character is not the most pertinent thing to ...

10

One important property of atom-centered basis sets is that electrons can only be localized on atoms. This is a problematic property when modeling solid systems with defects. For instance, at a color center, an electron is localized at a vacancy site. How can you model this with atom-centered basis sets? You have place a ghost atom at the vacancy site, ...

10

You might be interested to know that I've recently presented a general solution to this problem in Curing basis set overcompleteness with pivoted Cholesky decompositions, J. Chem. Phys. 151, 241102 (2019). The method is amazingly versatile, it also works if you have nuclei that are "unphysically" close to each other as I showed in Accurate reproduction of ...

10

Okay, so there are many layers to this question. cc-pVTZ for H is [5s2p1d/3s], which comes out to 3 + 2*3 + 5 = 3+6+5 = 14 basis functions per atom, which are composed of 16 primitives (the contracted s function). Now, while there are 1 and 3 cartesians for the s and p shells, for the d shell you have 6 cartesian functions but only 5 spherical functions. ...

10

The safe choice is to use diffuse functions on all atoms. It is quite rare to run into pathological overcompleteness with standard augmented basis sets, unless you're looking at very high-energy geometries. Overcompleteness may become an issue if you're using multiply augmented basis sets, such as d-aug-cc-pVXZ or t-aug-cc-pVXZ, but as I've recently shown in ...

10

In general, when computing any property with different models (e.g. level of theory, basis set, etc), if you don't have some kind of theoretical bound (like the variational principle) to determine what is a better result, you need a reference value to compare against. One choice for this reference is experimental results. At the end of the day, the goal of ...

9

The question was about "orbital basis sets" but explicitly mentions Gaussians; I guess the topic here is atomic orbital basis sets. In this case, the molecular orbitals are expanded as a linear combination of atomic orbitals (LCAO) as $\psi_i({\bf r}) = \sum_{\alpha} C_{\alpha i} \chi_\alpha({\bf r})$; minimizing the Hartree-Fock / density ...

9

In general, a system composed of $K$ interacting subsystems have a potential energy at a specific configuration of its parts. For instance, a system of $M$ nuclei and $N$ electrons can be separated into interacting subsystems with internal geometries, having $\{\mathbf{R}_A\}$ as nuclear positions for subsystem $A$ with $N_A$ electrons, $\{\mathbf{R}_B\}$ ...

9

I think a good way to see this is to simplify the stage a little. Imagine we want to solve the Schrödinger equation for a Hamiltonian $H$. For this we take a very simple basis, namely just two real-valued functions $\{f_1, f_2\}$. If we variationally optimise the trial wavefunction from this basis, we are presented with an approximation to the ground state ...

9

The Tran-Blaha functional is not a generally useful functional, so I would not describe it as "trustworthy" at all. For example, it is not size-extensive so twice as much material does not have twice the energy, and hence you can't use it for any thermodynamics or phase stability or finite-displacement phonons etc. There isn't a proper potential ...

9

There are many options to answer your question. Just to compare the computation time, the first condition is to run the simulations with the same hardware i.e. in the same system (two different set-ups even with the same type of hardware can perform differently). Also, both codes have to be compiled with the same compiler and with the same compiling ...

9

Your formula for the analytical $\ce{H}$ 1S orbital is incorrect. A 1S orbital has the form $$\phi_{1s}(\zeta_1,r)=\sqrt{\frac{\zeta_1^3}{\pi}}\exp(-\zeta_1 r)$$ For $\ce{H}$ the optimal value for $\zeta_1$ is $1.24$. If you update your plot in this way, you will see that the BSE plot is the correct one for $\ce{H}$. So what is wrong with the coefficients ...

8

I'm not completely satisfied with this answer, but I think it is worth including here what I have for now. Going back to the original papers that defined the various aug-cc-pVXZ basis sets, the even-tempered exponents of the diffuse functions are obtained by an optimization to fit CI calculations for those atoms. The functions $\{\zeta_i\}$ are obtained as \$...

8

The Karlsruhe def2 basis sets cover most of the periodic table and are based on effective core potentials for relativistic effects; these are a good starting point for whatever you are interested in, and are also available built-in in Gaussian. As to the functional, since you want to study slabs you probably want to use a pure functional since otherwise the ...

8

Integral Screening It is possible to place upper bounds on integrals based on the Cauchy-Schwartz inequality: $$\left( a a | b b \right) \ge \left( a c | b d \right)$$ If the first integral is computed and deemed insignificant, the latter is as well. One can further combine this with the appropriate density matrix element and decide that the resulting ...

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