Pure-state noninteracting $v$-representability of electron densities from Kohn-Sham calculations with finite basis sets

Within the linear combination of atomic orbitals (LCAO) approximation, one can distinguish two different Kohn-Sham potentials. One is the potential available numerically in calculations, and the other is the exact potential corresponding to the LCAO density. The latter is usually not available, but can be obtained from the total density by a numerical inversion procedure or, as is done here, analytically using only one LCAO Kohn-Sham orbital. In the complete basis-set limit, the lowest-lying Kohn-Sham orbital suffices to perform the analytical inversion, and the two potentials differ by no more than a constant. The relation between these two potentials is investigated here for diatomic molecules and several atomic basis sets of increasing size and quality. The differences between the two potentials are usually qualitative (wrong behavior at nuclear cusps and far from the molecule even if Slater-type orbitals are used) and $\delta$-like features at nodal planes of the lowest-lying LCAO Kohn-Sham orbital. Such nodes occur frequently in LCAO calculations and are not physical. Whereas the behavior of the potential can be systematically improved locally by the increase of the basis sets, the occurrence of nodes is not correlated with the size of the basis set. The presence of nodes in the lowest-lying LCAO orbital can be used to monitor whether the effective potential in LCAO Kohn-Sham equations can be interpreted as the potential needed for pure-state noninteracting $v$-representability of the LCAO density. Squares of such node-containing lowest-lying LCAO Kohn-Sham orbitals are nontrivial examples of two-electron densities which are not pure-state noninteracting $v$-representable.

Figure 1

$\Delta v\left(\mathbf{r}\right)$ along the bonding axis calculated with various STO basis sets for the ${\text{H}}_2$ molecule. The broken line indicates the value of $-{\varepsilon }_1$.

Figure 2

$\Delta v\left(\mathbf{r}\right)$ along the bonding axis calculated with various STO basis sets for the ${\text{N}}_2$ molecule. The broken line indicates the value of $-{\varepsilon }_1$.

Figure 3

$\Delta v\left(\mathbf{r}\right)$ along the bonding axis calculated with various STO basis sets for the LiH molecule. The broken line indicates the value of $-{\varepsilon }_1$.

Figure 4

$\Delta v\left(\mathbf{r}\right)$ along the bonding axis calculated with the QZ4P basis set for the ${\text{H}}_2$ molecule. The broken line indicates the value of $-{\varepsilon }_1$.

Figure 5

$\Delta v\left(\mathbf{r}\right)$ along the bonding axis calculated with the QZ4P basis set for the ${\text{N}}_2$ molecule. The broken line indicates the value of $-{\varepsilon }_1$.

Figure 6

$\Delta v\left(\mathbf{r}\right)$ along the bonding axis calculated with the QZ4P basis set for the LiH molecule. The broken line indicates the value of $-{\varepsilon }_1$.

Figure 7

Plot of $\Delta v\left(\mathbf{r}\right)$ for LiH, obtained from ${\phi }_2\left(\mathbf{r}\right)$ along bonding axis. The broken line indicates the value of $-{\varepsilon }_2$.