13
This sounds like you were exploring work at least related to the work by the Lilienfeld group equally hosting a dedicated site here about data sets already used in their earlier and ongoing exploration of chemical space, programs used to work with the data, and publications.
To go considerably higher in molecule count than QM9, you could either go for
GDB-...
10
It is perfectly fine, and actually quite common, to use big basis sets for the most important atoms (perhaps at the active site of the chemical reaction) and a smaller basis set for the surrounding hydrogen atoms (for example).
The only problem you might face would be, that this complicates the basis set extrapolation to get closer to the complete basis set ...
9
The ISOL24 database (http://www.thch.uni-bonn.de/tc.old/downloads/GMTKN/GMTKN55/ISOL24.html) contains molecules with up to 81 atoms!
The other answer says that there's a database called "OE" with molecules that have up to 174 atoms, but it is "not yet publicly available".
4
What we have to do here is a functional derivative.
Lets consider the following method of finding functional derivartives:
$$F[\rho+\delta \rho]-F[\rho] = \int_\Omega \delta\rho \frac{\delta F[\rho]}{\delta\rho}\mathrm{d}\mathbf{r}+\mathscr{O}(\delta\rho^2)\tag{1}\label{FuncDer}$$
Lets now consider the Thomas-Fermi kinetic energy functional using the above ...
3
As you have already stated in the question, (Ineq. 1) is the standard inequality used in Cauchy-Schwarz integral screening. (Ineq. 2) and (Ineq. 3) are not valid inequalities, which is clear if you select atomic orbitals such that a & b are spatially disjoint from c & d and the proposed upper bounds vanish. (Ineq. 4), (Ineq. 5), and (Ineq. 6) follow ...
2
The matrix $\mathbf{s}$ in your question are the eigenvalues of the overlap matrix $\mathbf{S}$, further the matrix $\mathbf{U}$ are a matrix of the eigenvector of the overlap matrix.
Now looking at the equation that gives you concern:
$$\mathbf{X} = \mathbf{U}\mathbf{s}^{-1/2}\mathbf{U}^\dagger$$
Since $\mathbf{s}$ is a diagonal matrix then taking the ...
2
The documentation for the Gaussian software is actually pretty complete, even if qualitative. On the subject of optimizations, I recommend you look up the freq and opt keywords.
The manual explains the procedure followed by the Berny optimization as involving construction of an initial analytical estimate of the Hessian using a simplified force field, with ...
2
Beyond other answers, I'd suggest the original PubChemQC project, which offers ~3 million molecules from PubChem optimized using DFT (B3LYP/6-31G*). Molecules include a wide variety of elements as long as the molecular mass is less than 500 Da. (Roughly speaking that should still handle ~38 carbon atoms.)
"PubChemQC Project: A Large-Scale First-Principles ...
1
Mulliken atomic charges can be defined as[1]:
$$q_A = Z_A-\sum_{\mu\in A}\left( \mathbf{P\cdot S} \right)_{\mu\mu} \tag{Szabo 3.196}$$
Here I have used the same notation as in Szabo[1], with $\mathbf{P}$ being the density matrix and $\mathbf{S}$ beign the overlap matrix. $Z$ is the nuclear charge, and using the greek letter $\mu$ as indicies indicates that ...
1
Diagonalization of a matrix means to find the eigenvalues of the matrix and put them into a diagonal matrix:
$$\mathbf{S}^\text{diag}=\mathrm{diag}\left( \lambda_1,...,\lambda_n \right)\tag{1}\label{diagmat}$$
for $\lambda_i$ being the eigenvalues of the matrix $\mathbf{S}$.
When we want to find the eigenvalues of the matrix $\mathbf{S}$ we solve an ...
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