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@ -803,11 +803,17 @@ In the present paper we have extended the recently proposed DFT-based basis set
We studied the H$_{10}$, C$_2$, N$_2$, O$_2$ and F$_2$ linear molecules up to full dissociation limits at near FCI level in increasing basis sets, and investigated how the basis set correction affect the convergence toward the CBS limits of the PES of these molecular systems.
The DFT-based basis set correction rely on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the regular coulomb interaction projected into an incomplete basis set $\basis$, ii) the fitting of such effective interaction with a long-range interaction used in RS-DFT, iii) the use of complementary correlation functional of RS-DFT.
In the present paper, we investigated points i) and iii) in order to to properly investigate atomization energies.
In this context, we propose a new scheme to design functionals fulfilling a) $S_z$ invariance, b) size extensivity. To achieve such requirements we proposed to use CASSCF wave functions leading to extensive energies, and to develop functionals using only $S_z$ invariant density-related quantities.
In the present paper, we investigated points i) and iii) in order to study atomization energies.
In this context, we proposed a new scheme to design functionals fulfilling a) $S_z$ invariance, b) size extensivity. To achieve such requirements we proposed to use CASSCF wave functions leading to extensive energies, and to develop functionals using only $S_z$ invariant density-related quantities.
The development of new $S_z$ invariant and size extensive functionals has lead us to investigate the role of two related quantities: the spin-polarization and the on-top pair density.
To achieve $S_z$ invariant in the context of DFT based on multi-configurational wave functions, an effective spin polarization depending on the total density and on-top pair density is commonly used. Nevertheless, such an effective spin density can be considered as \textit{ad hoc} as its expression is formally valid only for a single-determinant wave function and it can become complex for multi-configurational wave functions. Based on the previous work of some of the present authors, we use functionals depending \textit{explicitly} on the on-top pair density.
One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin-polarization without loos of accuracy.
From a practical point of view, this allows to remove the use of the effective spin-polarization\cite{PerSavBur-PRA-95} which has only a clear mathematical ground for single Slater determinant and can be become complex-valued in the case of multi-configurational wave functions. From a more fundamental aspect, this shows that the spin-polarization in DFT-related frameworks only mimic's the role of the on-top density.
Regarding the results of the present approach, the basis set correction systematically improves the near FCI calculation in a given basis set. More quantitatively, it is shown that the atomization energy $D_0$ is within the chemical accuracy for all systems but C$_2$ within a triple zeta quality basis set, whereas the near FCI values are far from that accuracy within the same basis set.
In the case of C$_2$, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis set correction.
Also, it is shown that the basis set correction gives substantial differential contribution along the PES only close to the equilibrium geometry, meaning that it cannot recover the dispersion forces missing because the incompleteness of the basis set. Although it can be looked as a failure of the basis set correction, in our context such behaviour is actually preferable as the dispersion forces are long-range effects and the present approach was designed to recover electronic correlation effects near the electron coalescence.
\bibliography{srDFT_SC}