15

Counterpoise correction is a method to limit an error that results when studying an intermolecular reaction using an incomplete basis set. Typically a basis set is not converged, and a calculation could always be improved with more basis functions. This is especially true for long range interactions, ie one often needs to add diffuse functions to the set ...


13

Inge Söderkvist (2009) has a nice write-up of solving the Rigid Body Movement Problem by singular value decomposition (SVD). Suppose we are given 3D points $\{x_1,\ldots,x_n\}$ that after perturbation take positions $\{y_1,\ldots,y_n\}$ respectively. We seek a rigid "motion", i.e. a rotation $R$ and translation $d$ combined, applied to points $x_i$ that ...


10

People use all sorts of bases in practice. For example, people use orthonormal bases in DG methods to ensure that the mass matrix in time stepping schemes is diagonal. People also use hierarchical bases when doing $p$ adaptivity because it makes the construction of constraints at faces where different polynomial degrees come together trivial. For higher ...


9

A counterpoise correction is an a posteriori correction that may be applied to correct for the basis set superposition error (BSSE). More specifically, it uses mixed basis sets with "Ghost orbitals". For more information see, Boys, S.F. and Bernardi, F., "The calculation of small molecular interactions by the differences of separate total energies. Some ...


9

This is a pretty big topic, but the basic important qualities of an atom-centered basis set are: The number of separate, unlinked functions it uses at the valence level (the so-called *zeta* count), which allow the modeling of that many different electronic environments, practically speaking. Additional higher angular momentum so-called *polarisation* ...


6

The nodal Lagrange bases are nice because they interpolate the functions at the knots: $$\phi_i(x_j) = \delta_{ij}$$ This means that you can read and plot solutions by looking only at the coefficients $u_j$ in the representation:$$u_h(x) = \sum_j u_j\phi_j(x)$$ That's much nicer than having to evaluate the sum at every point you care about knowing $u_h$.


6

This is actually quite simple. Let's say you have your (two-dimensional) reference triangle $\hat K=\left\{(\xi,\eta)\in {\mathbb R}^2: 0\le \xi \le 1, 0\le \eta\le 1, \xi \le (1-\eta)\right\}$. Then you know how to define the three basis functions on this triangle. They are: $$ \hat N_1(\xi,\eta) = 1-\xi-\eta,\\ \hat N_2(\xi,\eta) = \xi,\\ \hat N_3(\xi,\eta)...


5

The Gnu Scientific Library (GSL) has an implementation of B-splines, and the documentation can be found here. According to the documentation, GSL uses De Boor's algorithm, which is numerically stable.


5

While it doesn't answer your explict question, I offer the following important remark that might answer the intentions behind the question. The anomalies of the Gram-Scmidt process in case of dependence are avoided if one computes the orthogonalization instead by the standard Householder $QR$ factorization (or in the sparse case the Givens variant). The ...


4

If $v_2$ can be expressed as a linear combination of $\{ v_3, v_4, v_5 \}$, but not of a subset of these vectors, then, e.g., $v_5$ is in the linear span of $\{ v_2, v_3, v_4 \}$. So your algorithm would likely map $v_5$ onto $0$. Typically, if you apply the Gram-Schmidt proecess to the sequence of $v_i$, $1 \leq i \leq L$, a vector $v_k$ is mapped onto $0$ ...


4

Some research into the problem led me to two publications that do a fair job explaining the structure: Bank, Dupont, and Yserentant (1987) A book chapter by Pinksy, Malhotra, and Thompson (1994) (Page 183). Whereas [1] gives a fairly concise matrix structure of $\mathbf{S}_1$ (Eq. 5.3, two hierarchy levels only), it lacks a more detailed explanation. The ...


4

As a general rule, Slater basis elements are closer to the actual solution, and therefore fewer of them are needed. The advantage of a Gaussian basis is the Gaussian Product Theorem, which allows products of gaussians at different centers to be easily integrated. This allows for a significant speed up when attempting to converge around multiple nearby ...


4

In Engineering nodal bases are a good starting point for solid mechanics problems because the principle of virtual work for the discretised system $\mathbf{K} \mathbf{u} = \mathbf{f}$ reads $$ \delta \mathbf{u}^T \mathbf{K} \mathbf{u} = \delta \mathbf{u}^T \mathbf{f} $$ showing that $\mathbf{K} \mathbf{u}$ are indeed (equivalent) nodal internal forces, and $\...


4

I think you've got slightly the wrong end of the stick from the documentation. As with a lot of other software in the area, GMSH started out with low order, hard coded numberings. These are the ones with the ASCII art representations, which only give first and second order numberings for tetrahedra (hence there aren't any face nodes in the 4 node or 10 node "...


3

On quad/hex you can use tensor product polynomials. For example in 2d, you map the cell to a reference cell $[-1,1] \times [-1,1]$ and if $N$ is the degree, you would use $$ P_i(\xi) P_j(\eta), \qquad 0 \le i,j \le N $$ as the basis functions, where $\xi,\eta$ are coordinates on the reference cell and $P_i$ is the Legendre polynomial of degree $i$. These are ...


3

There are a variety of different bases in FEM, but most involve basis functions which are associated with topological entities, like vertices, edges, faces, and element interiors. This makes it possible to enforce various types of continuity by ensuring that degrees of freedom for such functions match at shared vertices/edges/faces. These basis functions ...


3

Wikipedia has an answer here: http://en.wikipedia.org/wiki/Basis_set_(chemistry)#Correlation-consistent_basis_sets Edit: adding introductory text from Wikipedia: Some of the most widely used basis sets are those developed by Dunning and coworkers, since they are designed to converge systematically to the complete-basis-set (CBS) limit using ...


3

The paper [1] gives an explicit construction of the Bernstein form of a set of orthogonal polynomials on simplices based on Legendre polynomials. [1] Farouki, R.T., Goodman, T.N.T and Sauer, T: Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains, Computer Aided Geometric Design 20 (2003), 209-230, DOI: 10....


3

Advantage of the Gaussian basis is that you can use the Gaussian Product Theorem to simplify the two electron integrals at different centers. However, Gaussians ($e^{-\alpha r^2}$) don't capture the cusp of the wavefunction nor the exponential decay ($e^{-r}$) naturally, so one needs more Gaussians for the given accuracy. Advantage of the Slater type ...


3

This link describes how to do this using P. Hall bases. On an only somewhat related note, if I were implementing this I would worry about the numerical instability of testing linear dependency. Make sure to use a method for testing independence of new matrices that allows for numerical inaccuracy - maybe comparing the norm of $A - p(A)$ to the norm of $A$, ...


2

Just compute a QR factorization of $B$. The last $p-n$ columns then have he required property of forming a basis complement. For large $n$ and small $p-n$, you can take a random $p\times k$ matrix $X$ with $k=p-n$ or slightly larger (for an increased likelihood of having numerical stability), and compute a QR factorization of $X-B(B^TX)$, which, with ...


2

In the standard form, KS-DFT is solved variationally, which means that additional degrees of freedom in the basis set must lead to a lower (or equal) energy. This is a very basic property of the variational method and the math is almost certainly explained on Wikipedia or equivalent. I am assuming the same functional is used. Each DFT functional provides ...


2

I think Rob Kirby told me once that he had written something on using Bernstein polynomials for FEM. Take a look at his web site at Texas Tech (or now at Baylor).


1

From @Wolfgang Bangerth's answer, there exists a mapping function $(x,y,z)=\Phi(\xi,\eta)$ which can be expressed as a function of the basis shape functions. Using a similar notation as @Wolfgang Bangerth, the reference or base triangle is defined as: $$ \left\{(\xi,\eta)\in {\mathbb R}^2: 0\le \xi \le 1, 0\le \eta\le 1, \xi \le (1-\eta)\right\} $$ The ...


1

I guess, the final result is suffering from instability (relative) of Gram-Schmidt, even in modified/stabilized forms. In this case, instead of going from the start: modifying the orthogonalization process, we can start with the final result instead. So, after solving a non-Hermitian eigenvalue problem (with a reliable algorithm), we obtained $X_L$ and $X_R$...


1

Density functional theory optimizes the total energy with respect to the (possibly constrained) electronic density. The usual numerical procedure for this optimization is embodied by the KS method. The KS procedure will give you the optimal total energy and the ground-state (gs) density for which this optimal energy is achieved. This optimization procedure ...


1

Not using pseudo-potentials is a great way to drive up the computational cost of a QMC enormously while not adding to the accuracy much at all. There's a very recent paper about PPs for QMC: http://jcp.aip.org/resource/1/jcpsa6/v139/i1/p014101_s1


1

Also take a look at John Burkhardt's libraries at http://people.sc.fsu.edu/~jburkardt/ His code is LGPL licensed, as opposed to the GPL used by the Gnu Scientific Library.


1

From a results point of view, for most things, it shouldn't really matter which you use - basis sets are generally constructed and labeled in such a way that for example a TZVP basis set will give comparable results whether it's using GTOs or STOs. The cases where they're more likely to give different results will be properties that depend heavily on non-...


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