Well-tempered gaussians

Huzinaga and co-workers have extended the even-tempered idea by developing a four-parameter function of the form  

0 and O one more set of diffuse f and g functions were added, for a total of 171 functions. 

Escf The self-consistent field energy in hartrees. 

S Sum of the squares of the reference configurations in the Cl wave function. This is a rough measure of the quality of the reference 

1 Cmax I The absolute value of the largest Cl coefficient appearing outside the reference mace. For multiple spin couplings C,nax I is taken as 

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Matsuoka, O. and Huzinaga, S., Relativistic well-tempered Gaussian basis sets, Chem. Phys. Lett. 140, 567-571 (1987). 

S. Huzinaga and M, Klobukowski, J. Mol. Struct. (Theochem), 167, 1 (1988). Well-Tempered Gaussian Basis Set Expansions of Roothaan-Hartree-Fock Atomic Wavefunctions for Lithium Through Mercury. 

Huzinaga S, Klobukowski M. Well-tempered Gaussian basis sets for the calculation of matrix Hartree—Fock wavefunctions. Chem Phys Lett. 1993 212(3-4) 260-4. 

The added bias potential, t), is added to the overall potential energy and is repulsive, Gaussian-shaped, and centered on the CV at the time of addition. This results in a net force that prevents the system from exploring previously visited states and instead encourages it to explore new regions of the CVs. To achieve smooth convergence of the bias potential, we use the well-tempered variant of metadynamics (WTM) [32] ... 

The tVi are the expansion coefficients for the M s-type Gaussians, and we can see immediately the link between Eq. [42] and the wave function quadrature. So, for the calculation of ASA-based promolecular electron densities, we first need to develop a scheme for the fitting of the atomic densities. The exponents of the Gaussians may be chosen from, e.g., a well-tempered series.The coefficients may then be fitted against the true atomic ab initio electron density. Once these exponents and coefficients are set, these Gaussian exponents and coefficients are universally applicable. Promolecular densities p (r) can then be obtained quickly from Eq. [41]. 

In both cases, an estimate of the free energy cannot be done with the usual protocol because changing Gaussian width corresponds to using variable-size bins in the estimation of a probability. The correct free energy can be retrieved through a Torrie-Valleau-like formula applied to the well-tempered case, when, for long simulation times, the bias increases constantly over the whole domain ... 

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. 

When developing the large basis sets which are required for the reliable calculation of van der Waals interactions, it is important to ensure that the basis set be constructed and extended in a systematic fashion. In recent work, Wells and Wilson have employed systematic sequences of even-tempered basis sets of Gaussian-type functions in conjunction with many-body perturbation theory calculations to study van der Waals interaction potentials, thereby ensuring basis set superposition errors and size-inconsistency problems are controlled. They used the Boys-Bernardi procedure as a test for the magnitude of the basis set superposition error rather than as a correction. 

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