Post-Hartree-Fock Atomic Kinetic Energies Data Glossary

Daniel Mejia Rodriguez dmejiarodriguez@ufl.edu Quantum Theory Project, Department of Physics, University of Florida, Gainesville, FL 32611 S.B. Trickey trickey@qtp.ufl.edu Quantum Theory Project, Departments of Physics and of Chemistry, University of Florida, Gainesville, FL 32611

(revised 23 Nov 2020; original 26 April 2018)

I Background

Very accurate Hartree-Fock (HF) wave functions IJQC.71.491 (1) for the first 86 neutral atoms were used to compute post-SCF kinetic energies for a variety of single-point kinetic energy density functionals (KEDFs). Those HF wave functions are significantly improved over the widely used Clementi & Roetti compilation ClementiRoetti (2). Orbital occupancies were obtained from the original dataset and degenerate orbitals were fractionally occupied.

Kinetic energies were obtained by numerical integration using a double exponential radial quadrature DoubleExponential (3) with $r_{min}=10^{-6}\;a_{0}$ , $r_{max}=50\;a_{0}$ and 200 points for all atoms. The numerical integration was confirmed to reproduce the first 15 digits of the Kohn-Sham non-interacting kinetic energy obtained by analytical integration over the semi-infinite $[0,\infty)$ interval.

II Notation

Hartree atomic units are used in all formulae. The notation of functions and constants is as follows:

•

$n=n(\mathbf{r})$ , the electron number density.
•

$c_{kf}=\big{(}3\pi^{2}\big{)}^{1/3}$ .
•

$c_{\textsc{tf}}=\frac{3}{10}c_{kf}^{2}$ , the Thomas-Fermi constant.
•

$s=|\nabla n|/\big{(}2c_{kf}n^{4/3}\big{)}$ , the reduced density gradient.
•

$p=\nabla^{2}n/\big{(}4c_{kf}^{2}n^{5/3}\big{)}$ , the reduced density Laplacian.
•

$\tilde{p}=\frac{\nabla n\cdot\nabla\nabla n\nabla\nabla n\cdot\nabla n}{16(3% \pi^{2})^{4/3}\lvert\nabla n\rvert^{2}n^{10/3}}$ . the reduced quadratic density Hessian.
•

$x=\frac{5}{27}s^{2}$ .
•

$b=2^{4/3}c_{kf}$ , a conversion factor.

In the summary below, analytic expressions for the non-interacting KEDFs $T_{s}[n]$ are given for the spin-unpolarized case ( $n_{\uparrow}=n_{\downarrow})$ . For spin-polarized electron densities, the corresponding expression $T_{s}[n_{\uparrow},n_{\downarrow}]$ can be obtained directly by applying the spin-scaling relation OliverPerdew (4)

T_{s}[n_{\uparrow},n_{\downarrow}]=\frac{1}{2}\Big{(}T_{s}[2n_{\uparrow}]+T_{s% }[2n_{\downarrow}]\Big{)}

(1)

The general form of a one-point KEDF is

T_{s}[n]=\int d\mathbf{r}\;t_{\textsc{tf}}F_{s}\big{(}n,\nabla n,\nabla^{2}n,% \ldots\big{)}

(2)

where $t_{\textsc{tf}}$ is the Thomas-Fermi kinetic energy density per unit volume (defined below), and $F_{s}$ is the enhancement function.

III Kinetic Energy Density Functionals

The header of each entry shows the notation used in the pop-up periodic table.

TF: Thomas-Fermi TF1 (5, 6)

t_{\textsc{tf}}=c_{\textsc{tf}}n^{5/3}\;\;;\;\;F_{s}=1

(3)

W: von Weizsäcker W (7)

F_{s}^{\textsc{w}}(s)=\tfrac{5}{3}s^{2}

(4)

GEA2: Second-order gradient expansion GEA2_1 (8, 9)

F_{s}^{\textsc{gea2}}(s,p)=1+\tfrac{5}{27}s^{2}+\tfrac{20}{9}p

(5)

GEA4: Fourth-order gradient expansion GEA4 (10)

F_{s}^{\textsc{gea4}}(s,p)=1+\tfrac{5}{27}s^{2}+\tfrac{20}{9}p+\tfrac{8}{81}p^% {2}-\tfrac{1}{9}s^{2}p+\tfrac{8}{243}s^{4}

(6)

TFW: Thomas-Fermi plus von Weizsäcker

F_{s}^{\textsc{tfw}}(s)=1+\tfrac{5}{3}s^{2}

(7)

TF5W: Thomas-Fermi plus $\frac{1}{5}$ von Weizsäcker

F_{s}^{\textsc{tf5w}}(s)=1+\tfrac{1}{3}s^{2}

(8)

P82: Pearson P82 (11)

F_{s}^{\textsc{p82}}(s)=1+\frac{5}{27}\frac{s^{2}}{1+s^{6}}

(9)

DK87: DePristo and Kress DK87 (12)

F_{s}^{\textsc{dk86}}(x)=\frac{1+0.95x+14.281111x^{2}-19.57962x^{3}+26.64765x^% {4}}{1-0.05x+9.99802x^{2}+2.96805x^{3}}

(10)

LLP: Lee, Lee, and Parr LLP (13)

F_{s}^{\textsc{llp}}(s)=1+\frac{0.0044188b^{2}s^{2}}{1+0.0253bs\;\sinh^{-1}(bs)}

(11)

OL1 and OL2: Ou-Yang and Levy OL (14)

F_{s}^{\textsc{ol1}}(s)=1+\frac{5}{27}s^{2}+0.01354\frac{c_{kf}}{c_{\textsc{tf% }}}s

(12)

F_{s}^{\textsc{ol2}}(s)=1+\frac{5}{27}s^{2}+\frac{0.1774}{c_{\textsc{tf}}}\;% \frac{c_{kf}s}{1+8c_{kf}s}

(13)

P92: Perdew P92 (15)

F_{s}^{\textsc{p92}}(s)=\frac{1+88.396s^{2}+16.3683s^{4}}{1+88.2108s^{2}}

(14)

T92: Thakkar T92 (16)

F_{s}^{\textsc{t92}}(s)=1+\frac{0.0055b^{2}s^{2}}{1+0.0253\sinh^{-1}(bs)}-% \frac{0.072bs}{1+2^{5/3}bs}

(15)

LC94: Lembarki and Chermette LC94 (17)

F_{s}^{\textsc{lc94}}(s)=\frac{1+0.093907s\sinh^{-1}(76.32s)+0.26608s^{2}-0.08% 09615s^{2}e^{-100s^{2}}}{1+0.093907s\sinh^{-1}(76.32s)+0.000057767s^{4}}

(16)

LP97: Liu and Parr LP97 (18)

F_{s}^{\textsc{lp97}}(n)=\frac{3.26422}{c_{\textsc{tf}}}-\frac{0.02631}{c_{% \textsc{tf}}}\left(\int d\mathbf{r}\;n^{4/3}\right)n^{-1/3}+\frac{0.000498}{c_% {\textsc{tf}}}\left(\int d\mathbf{r}\;n^{11/9}\right)^{2}n^{-4/9}

(17)

VJKS00: Vitos, Johansson, Kollár, and Skriver VJKS00 (19)

F_{s}^{\textsc{vjk00}}(s)=\frac{1+0.8944s^{2}-0.0431s^{6}}{1+0.6511s^{2}+0.043% 1s^{4}}

(18)

E00: Ernzerhof E00 (20)

F_{s}^{\textsc{e00}}(s)=\frac{135+28s^{2}+5s^{4}}{135+3s^{2}}

(19)

TW02: Tran and Wesolowski TW02 (21)

F_{s}^{\textsc{tw02}}(s)=1+0.8438-\frac{0.8438}{1+0.27482816s^{2}}

(20)

PBE2 and PBE4: Karasiev, Trickey, and Harris PBEn (22)

F_{s}^{\textsc{pbe2}}(s)=1+2.0309\frac{s^{2}}{1+0.2942s^{2}}

(21)

F_{s}^{\textsc{pbe4}}(s)=1-7.2333\frac{s^{2}}{1+1.7107s^{2}}+61.645\Big{(}% \frac{s^{2}}{1+1.7107s^{2}}\Big{)}^{2}-93.683\Big{(}\frac{s^{2}}{1+1.7107s^{2}% }\Big{)}^{3}

(22)

MGGA: Perdew and Constantin MGGA (23)

F_{s}^{\textsc{mgga}}(s,p)=F_{s}^{\textsc{w}}(s)+z(s,p)f\big{(}z(s,p)\big{)}

(23)

z(s,p)=\frac{F_{s}^{\textsc{gea4}}(s,p)}{\sqrt{1+\left(\Delta/(1+F_{s}^{% \textsc{w}})\right)^{2}}}-F_{s}^{\textsc{w}}(s)

(24)

\Delta=\tfrac{8}{81}p^{2}-\tfrac{1}{9}s^{2}p+\tfrac{8}{243}s^{4}

(25)

f(z)=\begin{cases}0,&z\leqslant 0.5389\\ \left[\frac{1+e^{0.5389/(0.5389-z)}}{e^{0.5389/z}+e^{0.5389/(0.5389-z)}}\right% ]^{3},&0<z<0.5389\\ 1,&z\geqslant 0.5389\end{cases}

(26)

GDS08: Ghiringhelli and Delle Site GDS08 (24)

F_{s}^{\textsc{gds08}}(n,s)=\frac{5}{3}s^{2}+\frac{0.860}{c_{\textsc{tf}}n^{2/% 3}}+\frac{0.224\ln(n)}{c_{\textsc{tf}}n^{2/3}}

(27)

RDA: Karasiev, Jones, Trickey, and Harris RDA (25)

F_{s}^{\textsc{rda}}(s,p)=\frac{5}{3}s^{2}+0.50616+3.04121\Big{(}\frac{\tilde{% \kappa}_{4a}}{1+1.29691\tilde{\kappa}_{4a}}\Big{)}^{2}-0.34567\Big{(}\frac{% \tilde{\kappa}_{4b}}{1+0.56184\tilde{\kappa}_{4b}}\Big{)}^{4}-1.89738\frac{% \kappa_{2c}}{1+0.21944\kappa_{2c}}

(28)

\tilde{\kappa}_{4a}=\sqrt{s^{4}+46.47662p^{2}}

(29)

\tilde{\kappa}_{4b}=\sqrt{s^{4}+18.80658p^{2}}

(30)

\kappa_{2c}=s^{2}-0.90346p

(31)

APBEK and revAPBEK: Constantin, Fabiano, Laricchia, and Della Sala APBEK (26)

F_{s}^{\textsc{apbek}}(s)=1+\frac{0.23889s^{2}}{1+\tfrac{0.23889}{0.804}s^{2}}

(32)

F_{s}^{\textsc{revapbek}}(s)=1+\frac{0.23889s^{2}}{1+\tfrac{0.23889}{1.245}s^{% 2}}

(33)

VT84F: Karasiev, Chakraborty, Shukruto, and Trickey VT84F (27)

F_{s}^{\textsc{vt84f}}(s)=1-\frac{\mu s^{2}e^{-\alpha s^{2}}}{1+\mu s^{2}}+% \Big{(}1-e^{-\alpha s^{4}}\Big{)}\Big{(}s^{-2}-1\Big{)}+\frac{5}{3}s^{2}

(34)

\mu=2.777028126\;\;\;\;;\;\;\;\;\alpha=\mu-\tfrac{40}{27}

(35)

L04 and L06: Laricchia, Constantin, Fabiano, and Della Sala Lk (28)

F_{s}^{\textsc{lk}}(s,p)=1+2\kappa-\Big{(}\frac{\kappa}{1+x_{1}/\kappa}+\frac{% \kappa}{1+x_{2}/\kappa}\Big{)}

(36)

x_{1}=x+\Delta+\frac{x^{2}}{\kappa}

(37)

x_{2}=2\frac{x\Delta}{\kappa}+\frac{x^{3}}{\kappa^{2}}

(38)

\Delta=\tfrac{8}{81}p^{2}-\tfrac{1}{9}s^{2}p+\tfrac{8}{243}s^{4}

(39)

with $\kappa=0.402$ (L0.4) or $\kappa=0.6225$ (L0.6)

revMGGA and revMGGAloc: Cancio, Stewart, and Kuna revMGGA (29); Cancio and Redd revMGGAloc (30)

Note (23 Nov. 2020): In Cancio and Redd revMGGAloc (30) these are called mGGArev4 and mGGAloc4 respectively. There is no explicit equation for mGGAloc4 there but it can be ascertained by examining their Eq. (37), which gives their “local GEA”, including the coefficients listed below. That in combination with their Eqs. (20) and (21) gives the mGGAloc4 form shown here. It also is described briefly in their section 4.5. (see also Figure 8 and Table 2): “the short-dashed and dot-dashed lines show the mGGAloc with $\alpha=1$ and $\alpha=4$ , which adhere to the local GEA outside the transition region.”

F_{s}^{\textsc{revmgga}}(s,p)=1+F_{s}^{\textsc{w}}(s)+z_{1}(s,p)f\big{(}z_{1}(% s,p)\big{)}

(40)

F_{s}^{\textsc{revmggaloc}}(s,p)=1+F_{s}^{\textsc{w}}(s)+z_{2}(s,p)f\big{(}z_{% 2}(s,p)\big{)}

(41)

z_{1}(s,p)=F_{s}^{\textsc{gea2}}(s,p)-\left(1+F_{s}^{\textsc{w}}(s)\right)

(42)

z_{2}(s,p)=1-0.275s^{2}+2.895p-\left(1+F_{s}^{\textsc{w}}(s)\right)

(43)

f(z)=\left[1-e^{-1/|z|^{4}}\big{(}1-H(z)\big{)}\right]^{1/4}

(44)

where $H(z)$ is the Heaviside unit step function.

TFLreg: Regularized Thomas-Fermi plus Laplacian TFLreg (31)

F_{s}^{\textsc{tflreg}}(s,p)=\mathrm{Max}\big{(}1+\tfrac{20}{9}p\,,\,\tfrac{5}% {3}s^{2}\big{)}

(45)

LGAPGE and LGAP: Constantin, Fabiano, Śmiga, and Della Sala LGAP (32)

F_{s}^{\textsc{lgapge}}(s)=1+0.0131s+0.18528s^{2}+0.0262s^{3}

(46)

F_{s}^{\textsc{lgap}}(s)=1+0.8\Big{(}1-e^{-\mu_{1}s-\mu_{2}s^{2}-\mu_{3}s^{3}}% \Big{)}

(47)

\mu_{1}=\tfrac{0.0131}{0.8}

(48)

\mu_{2}=\tfrac{0.18528}{0.8}+\tfrac{\mu_{1}^{2}}{2}

(49)

\mu_{3}=\tfrac{0.0262}{0.8}+\mu_{1}\mu_{2}-\tfrac{\mu_{1}^{3}}{6}

(50)

MVT84F: Mejia-Rodriguez and Trickey MVT84F (33)

F_{s}^{\textsc{mvt}}[s]=\theta_{\textsc{mvt}}[n]F_{s}^{\textsc{vt84F}}[n]+% \left(1-\theta_{\textsc{mvt}}[n]\right)F_{s}^{\textsc{w}}[n]

(51)

\theta_{\textsc{mvt}}=\mathrm{Erf}\left[\sqrt{\Theta}\right]

(52)

with the DORI JCTC.10.3745 (34)

\Theta:=4\left(1+\frac{\tilde{p}}{s^{4}}-2\frac{p}{s^{2}}\right)

(53)

LKT: Luo, Karasiev, and Trickey LKT (35)

F_{s}^{\textsc{lkt}}[s]=\frac{1}{\cosh(as)}+F_{s}^{\textsc{w}}[n]

(54)

a=1.3

(55)

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