Numerical calculations of the linear-response functions and the Fukui functions of a fractional system. All calculations are performed in the level B3LYP/6-31G(d) in QM4D package. 59 λ’s in subfigures (a) and (b) indicate the stepsize for the numerical derivatives. The path of the plot is along the z-axis for both subfigures. (a) The linear-response function of Li0.2 + with N α = 1.8 and N β = 1.0. A δ potential at R = (0.0, 0.0, 1.0) au is set to evaluate the numerical derivative. (b) The α-spin Fukui function f α(r) = ∂ρ(r)/∂N α of a special lithium atom with 1.8 α electrons and 1.2 β electrons.
Numerical and analytical p + q = 3 derivatives for CO molecule at the level of LDA/6-31G(d). The plot path is along the z-axis. The coordinates of the carbon and the oxygen atoms are (0.0, 0.0, 2.0) and (0.0, 0.0, 4.104777) a.u., respectively. λ indicates the stepsize of the numerical derivatives for all cases. When N derivative is concerned, only electron removal direction is considered.
δ p + q E/δN p δv q in the Taylor expansion of E[N, v] up to the order of p + q = 3. See the text for the meaning of each derivative.
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