Leading corrections to local approximations. II. The case with turning points

Quantum corrections to Thomas-Fermi (TF) theory are investigated for noninteracting one-dimensional fermions with known uniform semiclassical approximations to the density and kinetic energy. Their structure is analyzed, and contributions from distinct phase space regions (classically-allowed versus forbidden at the Fermi energy) are derived analytically. Universal formulas are derived for both particle numbers and energy components in each region. For example, in the semiclassical limit, exactly ${\left(6\pi \sqrt{3}\right)}^{-1}$ of a particle leaks into the evanescent region beyond a turning point. The correct normalization of semiclassical densities is proven analytically in the semiclassical limit. Energies and densities are tested numerically in a variety of one-dimensional potentials, especially in the limit where TF theory becomes exact. The subtle relation between the pointwise accuracy of the semiclassical approximation and integrated expectation values is explored. The limitations of the semiclassical formulas are also investigated when the potential varies too rapidly. The approximations are shown to work for multiple wells, except right at the mid-phase point of the evanescent regions. The implications for density functional approximations are discussed.

Figure 1

$\gamma n_{\gamma }\left(x\right)$ for $N=1$ harmonic potential where $\gamma =1$ (blue, with semiclassical approximation dashed red), 1/4 (orange), 1/16 (green), and TF (black).

Figure 2

Classical properties of ${\epsilon }_F$ trajectory for the SHO with $N=1$ vs $x-x_F$. Red is ${\epsilon }_F-v\left(x\right)$, dark red is $k_F\left(x\right)$, blue is ${\theta }_F\left(x\right)$, and dark blue is $d_F\left(x\right)$.

Figure 3

$K_i$ functions versus $\theta$, with $K_i$ versus $-i\theta$ in the evanescent region. $K_0$ is light blue, $K_1$ is red, $K_2$ is black, and the zero function is dark blue.

Figure 4

Same as Fig. 1, but for the kinetic energy density.

Figure 5

$n^{\text{sc}}\left(x\right)/n\left(x\right)$ for the harmonic oscillator with $\gamma =1$ (blue), 1/2 (red), 1/4 (green), 1/8 (orange), and 1/16 (black).

Figure 6

Analytic potentials employed for the systematic investigation of the uniform semiclassical approximations.

Figure 7

Scaled dimensionless turning-point density, ${\gamma }^{2/3}{l}_{F}n\left(x_F\right)$, as a function of $\gamma$ for several potentials. Points are exact, and straight lines are the leading behavior of the semiclassical formula, Eq. (21). Black is the harmonic oscillator, and blue and red are the left and right turning points of a Morse potential of depth 8 ($a=1/2$).

Figure 8

Scaled dimensionless kinetic energy turning-point density $l_{\mathrm{F}}^3{t}_{\gamma }\left(x_{\mathrm{F}}\right)$. Points are exact, straight lines are fits to the exact except for the horizontal which is the semiclassical result. The colors refer to the same potentials as in Fig. 7.

Figure 9

$N_{\gamma }^{\mathrm{forbid}}$ for harmonic oscillator (black) and left- (blue) and right- (red) turning points of the Morse potential (all with $N=1$). The straight lines are the leading asymptotic behavior as $\gamma \to 0$, given by Eq. (38). The empty circles correspond to results obtained with Eq. (21), while the filled are exact.

Figure 10

Same as Fig. 9, except the deviation from $N/\gamma$ in the allowed region.

Figure 11

Same as Fig. 9, but for $V_{\gamma }^{\mathrm{forbid}}/\left|{\epsilon }_F\right|$.

Figure 12

Same as Fig. 11 but for the allowed region.

Figure 13

Same as Fig. 9 but for $l_F^2{\gamma }^{-2/3}{T}_{\gamma }^{\mathrm{forbid}}$.

Figure 14

Same as Fig. 13 but for the allowed region.

Figure 15

$V-V^{TF}$, blue = SHO, green = quartic, red = PT, purple = Morse; empty = sc, filled = exact (when filled is not shown it is because it is zero).

Figure 16

$T-T^{TF}$, blue = SHO, green = quartic, red = PT, purple = Morse; empty = sc, filled = exact (when filled is not shown it is because it is zero).

Figure 17

Densities for PT with $D=1/2+1/8$. Blue is exact, black is TF, red is semiclassical, while dark red is semiclassical with just the first contribution.

Figure 18

Same as Fig. 17 but with $D=1/2+1/64$.

Figure 19

Double well potentials (blue) with near degenerate eigenstates (black) and Fermi level (red) for $b=3.0$ in Eq. (48) and for $b=3.3$.

Figure 20

Particle density (blue is exact, red is semiclassical, black is TF) for the double well of Fig. 19 with $b=3.3$.

Figure 21

Same as Fig. 20 but for kinetic energy density.

Figure 22

Same as Fig. 20 but for $b=3.0$.

Figure 23

Same as Fig. 21 but for $b=3.0$.

Figure 24

Positive kinetic energy $t^p\left(x\right)$: Exact (blue), semiclassical (dot-dashed red), semiclassical without higher-order terms in $\gamma$ (continuous red).