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17.4 Density Functionals

In the following, $\rho_\alpha$ and $\rho_\beta$ are the $\alpha$ and $\beta$ spin densities; the total spin density is $\rho$;

The gradients of the density enter through

$$\sigma_{\alpha\alpha} = \nabla\rho_\alpha \cdot \nabla\rho_\alpha \; , \sigma_{\beta\beta... ...\; , \sigma = \sigma_{\alpha\alpha}+\sigma_{\beta\beta}+2\sigma _{\alpha\beta}.$$ (5)

$$\chi_\alpha = \frac{\sqrt{\sigma_{\alpha\alpha}}}{\rho_\alpha^{4/3}}\;, \chi_\beta = \frac{\sqrt{\sigma_{\beta\beta}}}{\rho_\beta^{4/3}}\;.$$ (6)

$$\upsilon_\alpha = \nabla^2\rho_\alpha \; , \upsilon_\beta=\nabla^2\rho_\beta \; , \upsilon=\upsilon_\alpha+\upsilon_\beta \;.$$ (7)

Additionally, the kinetic energy density for a set of (Kohn-Sham) orbitals generating the density can be introduced through

$$\tau_\alpha = \sum_i^\alpha \left\vert{\bf\nabla}\phi_i\right\vert^2 \; , \tau_... ...ta \left\vert{\bf\nabla}\phi_i\right\vert^2 \;, \tau=\tau_\alpha+\tau_\beta \;.$$ (8)

All of the available functionals are of the general form

$$F\left[\rho_s,\rho_{\bar{s}}, \sigma_{ss},\sigma_{\bar{s}\bar{s}},\sigma_{s\bar{s}}, \tau_s,\tau_{\bar{s}}, \upsilon_s,\upsilon_{\bar{s}} \right] = \int d^3{\bf r} K\left(\rho_s,\rho_{\bar{s}}, \sigma_{ss},\sigma_... ...\sigma_{s\bar{s}}, \tau_s,\tau_{\bar{s}}, \upsilon_s,\upsilon_{\bar{s}} \right)$$ (9)

where $\bar{s}$ is the conjugate spin to $s$.

Below is a list of keywords for the functionals supported by MOLPRO. Additionally there are a list of alias keywords deatailed in the next section for various combinations of the primary functionals listed below.

PBEC

PBE Correlation Functional
doi:10.1103/PhysRevLett.77.3865

PW86

.. GGA Exchange Functional.
doi:10.1103/PhysRevB.33.8800

B95

Becke 1995 Correlation Functional. $\\ tau$ dependent Dynamical correlation functional.
doi:10.1063/1.470829

TH4

.. Density an gradient dependent first and second row exchange-correlation functional.
doi:TH3/4

TH2

.. Density and gradient dependent first row exchange-correlation functional.
doi:10.1021/jp980259s

ECERF

Short-range LDA correlation functional. Local-density approximation of correlation energy
for short-range interelectronic interaction ${\rm erf}(\mu r_{21})/r_{12}$,
S. Paziani, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Phys. Rev. B 73, 155111 (2006).

$$\nonumber \epsilon_c^{\rm SR}(r_s,\zeta,\mu) =\epsilon_c^{\rm... ...^3+a_2 \mu^4+ a_3\mu^5+a_4\mu^6+a_5\mu^8}{(1+b_0^2\mu^2)^4},$$

where

$$Q(x)=\frac{2\ln(2)-2}{\pi^2}\ln\left(\frac{1+a\,x+b\,x^2+c\,x^3}{1+a\,x+ d\,x^2}\right),$$ (10)

with $a=5.84605$, $c=3.91744$, $d=3.44851$, and $b=d-3\pi\alpha/(4\ln(2)-4)$. The parameters $a_i(r_s,\zeta)$ are given by

$$a_1 = 4 \,b_0^6 \,C_3+b_0^8 \,C_5,$$

$$a_2 = 4 \,b_0^6 \,C_2+b_0^8\, C_4+6\, b_0^4 \epsilon_c^{\rm PW92},$$

$$a_3 = b_0^8 \,C_3,$$

$$a_4 = b_0^8 \,C_2+4 \,b_0^6\, \epsilon_c^{\rm PW92} ,$$

$$a_5 = b_0^8\,\epsilon_c^{\rm PW92},$$

with

$$C_2 = -\frac{3(1\!-\!\zeta^2)\,g_c(0,r_s,\zeta\!=\!0)}{8\,r_s^3}$$

$$C_3 = - (1\!-\!\zeta^2)\frac{g(0,r_s,\zeta\!=\!0)}{\sqrt{2\pi}\, r_s^3}$$

$$C_4 = -\frac{9\, c_4(r_s,\zeta)}{64 r_s^3}$$

$$C_5 = -\frac{9\, c_5(r_s,\zeta)}{40\sqrt{2 \pi} r_s^3}$$

$$c_4(r_s,\zeta) = \left(\frac{1\!+\!\zeta}{2}\right)^2g''\left(0, r_s\left(\frac{2}... ...\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ \left(\frac{1\!-\!\zeta}{2}\right)^2 \times$$

$$g''\left(0, r_s\left(\frac{2}{1\!-\!\zeta}\right)^{1/3}\!\!\!\!\!... ...!=\!1\right) + (1\!-\!\zeta^2)D_2(r_s)-\frac{\phi_8(\zeta)}{5\,\alpha^2\,r_s^2}$$

$$c_5(r_s,\zeta) = \left(\frac{1\!+\!\zeta}{2}\right)^2g''\left(0, r_s\left(\frac{2}... ...\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ \left(\frac{1\!-\!\zeta}{2}\right)^2 \times$$

$$g''\left(0, r_s\left(\frac{2}{1\!-\!\zeta}\right)^{1/3}\!\!\!\!\!\!\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ (1\!-\!\zeta^2)D_3(r_s),$$ (11)

and

$$\phantom{\bigl[} b_0(r_s) = 0.784949\,r_s$$ (12)

$$\phantom{\Biggl[} g''(0,r_s,\zeta\!=\!1) = \frac{2^{5/3}}{5\,\alpha^2 \,r_s^2} \, \frac{1-0.02267 r_s}{\left(1+0.4319 r_s+0.04 r_s^2\right)}$$ (13)

$$\phantom{\Biggl[}D_2(r_s) = \frac{e^{- 0.547 r_s}}{r_s^2}\left(-0.388 r_s+0.676 r_s^2\right)$$ (14)

$$\phantom{\Biggl[}D_3(r_s) = \frac{e^{-0.31 r_s}}{r_s^3}\left(-4.95 r_s+ r_s^2\right).$$ (15)

Finally, $\epsilon_c^{\rm PW92}(r_s,\zeta)$ is the Perdew-Wang parametrization of the correlation energy of the standard uniform electron gas [J.P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992)], and

$$g(0,r_s,\zeta\!=\!0)=\frac{1}{2}(1-Br_s+Cr_s^2+Dr_s^3+Er_s^4)\,{\rm e}^{-dr_s},$$ (16)

is the on-top pair-distribution function of the standard jellium model [P. Gori-Giorgi and J.P. Perdew, Phys. Rev. B 64, 155102 (2001)], where $B=-0.0207$, $C=0.08193$, $D=-0.01277$, $E=0.001859$, $d=0.7524$. The correlation part of the on-top pair-distribution function is $g_c(0,r_s,\zeta\!=\!0)=g(0,r_s,\zeta\!=\!0)-\frac{1}{2}$.

B97RDF

Density functional part of B97 Re-parameterized by Hamprecht et al. Re-parameterization of the B97 functional in a self-consistent procedure by Hamprecht et al. This functional needs to be mixed with 0.21*exact exchange.
doi:10.1063/1.477267

PBEX

PBE Exchange Functional
doi:10.1103/PhysRevLett.77.3865

HCTH147

Handy least squares fitted functional
doi:10.1063/1.480732

THGFC

.. Density and gradient dependent first row exchange-correlation functional for closed shell systems. Total energies are improved by adding $DN$, where $N$ is the number of electrons and $D=0.1863$.
doi:10.1016/S0009-2614(97)00586-1

B86

X $\alpha\beta\gamma$. Divergence free semiempirical gradient-corrected exchange energy functional. $\lambda=\gamma$ in ref.
doi:10.1063/1.450025

BW

Becke-Wigner Exchange-Correlation Functional. Hybrid exchange-correlation functional comprimising Becke's 1998 exchange and Wigner's spin-polarised correlation functionals.
doi:10.1039/FT9959104337

PW92C

Perdew-Wang 1992 GGA Correlation Functional. Electron-gas correlation energy.
doi:10.1103/PhysRevB.45.13244

BR

Becke-Roussel Exchange Functional. A. D. Becke and M. R. Roussel,Phys. Rev. A 39, 3761 (1989)

$$K=\frac{1}{2}\sum_s \rho_s U_s ,$$ (17)

where

$$U_s=-(1-e^{-x}-xe^{-x}/2)/b ,$$ (18)

$$b=\frac{x^3e^{-x}}{8\pi\rho_s}$$ (19)

and $x$ is defined by the nonlinear equation

$$\frac{xe^{-2x/3}}{x-2}=\frac{2\pi^{2/3}\rho_s^{5/3}}{3Q_s} ,$$ (20)

where

$$Q_s=(\upsilon_s-2\gamma D_s)/6 ,$$ (21)

$$D_s=\tau_s-\frac{\sigma_{ss}}{4\rho_s}$$ (22)

and

$$\gamma=1.$$ (23)

B88C

Becke 1988 Correlation Functional. Correlation functional depending on B86MGC exchange functional with empirical atomic parameters, $t$ and $u$. The exchange functional that is used in conjunction with B88C should replace B88MGC here.
doi:10.1063/1.454274

P86

.. Gradient correction to VWN.
doi:10.1103/PhysRevB.33.8822

TH3

.. Density and gradient dependent first and second row exchange-correlation functional.
doi:TH3/4

B86MGC

X $\alpha\beta\gamma$ with Modified Gradient Correction. B86 with modified gradient correction for large density gradients.
doi:10.1063/1.451353

MK00

Exchange Functional for Accurate Virtual Orbital Energies
doi:10.1063/1.481298

PW91C

Perdew-Wang 1991 GGA Correlation Functional
doi:10.1103/PhysRevB.46.6671

HCTH120

Handy least squares fitted functional
doi:10.1063/1.480732

PW91X

Perdew-Wang 1991 GGA Exchange Functional
doi:10.1103/PhysRevB.46.6671

TH1

Tozer and Handy 1998. Density and gradient dependent first row exchange-correlation functional.
doi:10.1063/1.475638

LTA

Local $\\ tau$ Approximation. LSDA exchange functional with density represented as a function of $\\ tau$.
doi:10.1063/1.479374

CS2

Colle-Salvetti correlation functional. R. Colle and O. Salvetti, Theor. Chim. Acta 37, 329 (1974); C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37, 785(1988)

CS2 is defined through

$$K = -a \left({ \rho+2b\rho^{-5/3} \left[ \rho_\alpha t_{\alpha} + \rh... ...ta t_{\beta} -\rho t_W \right] e^{-c\rho^{-1/3}} \over 1+d \rho^{-1/3} }\right)$$ (24)

where

$$t_{\alpha} = \frac{\tau_\alpha}{2}-\frac{\upsilon_\alpha}{8}$$ (25)

$$t_{\beta} = \frac{\tau_\beta}{2}-\frac{\upsilon_\beta}{8}$$ (26)

$$t_{W} = {1\over 8} {\sigma \over \rho} - {1\over 2} \upsilon$$ (27)

and the constants are $a=0.04918, b=0.132, c=0.2533, d=0.349$.

PBEXREV

Revised PBE Exchange Functional. Changes the value of the constant R from the original PBEX functional
doi:10.1103/PhysRevLett.80.890

EXERF

Short-range LDA correlation functional. Local-density approximation of exchange energy
for short-range interelectronic interaction ${\rm erf}(\mu r_{12})/r_{12}$,
A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996).

$$\epsilon_x^{\rm SR}(r_s,\zeta,\mu) = \frac{3}{4\pi}\frac{\ph... ...a)^{4/3} f_x\left(r_s,\mu(1\!-\!\zeta)^{-1/3}\right) \nonumber$$

with

$$\phi_n(\zeta)=\frac{1}{2}\left[ (1\!+\!\zeta)^{n/3}+(1\!-\!\zeta)^{n/3} \right],$$ (28)

$$f_x(r_s,\mu) = -\frac{\mu}{\pi}\biggl[(2y-4y^3)\,e^{-1/4y^2}... ...rac{1}{2y}\right)\biggr], \qquad y=\frac{\mu\,\alpha\,r_s}{2},$$ (29)

and $\alpha=(4/9\pi)^{1/3}$.

THGFCFO

.. Density and gradient dependent first row exchange-correlation functional. FCFO = FC + open shell fitting.
doi:10.1016/S0009-2614(97)00586-1

STEST

Test for number of electrons

LYP

Lee, Yang and Parr Correlation Functional. C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37, 785(1988); B. Miehlich, A. Savin, H. Stoll and H. Preuss, Chem. Phys. Letters 157, 200 (1989)

$$K = 4\,{\frac {A\rho_{\alpha}\rho_{\beta}Z}{\rho}}+AB\omega\sigma\left ( \rho_{\alpha}\rho_{\beta}\left (47-7\,\delta\right )/18-2{\rho}^ {2}/3\right )$$

$$+ \sum_s AB\omega\,\biggl (\rho_{s}\rho_{\bar s}\left (8\,{2}^{2/3}... ...s}- {\frac {\left (\delta-11\right )\rho_{s}\sigma_{ss}}{9\rho}}\right)\biggr..$$

$$\biggl.+ \left (2{\rho}^{2}/3-\rho_{s}^{2}\right ) \sigma_{\bar s \bar s}\biggr ) ,$$ (30)

where

$$\omega={e^{-{\frac {c}{{\rho}^{1/3}}}}}Z{\rho}^{-11/3} ,$$ (31)

$$\delta={\frac {c+dZ}{{\rho}^{1/3}}} ,$$ (32)

$$B= 0.04918 ,$$ (33)

$$A= 0.132 ,$$ (34)

$$c= 0.2533 ,$$ (35)

$$d= 0.349 ,$$ (36)

$$e=\frac{3}{10}\left (3\pi^2\right )^{2/3}$$ (37)

and

$$Z=\left (1+{\frac {d}{{\rho}^{1/3}}}\right )^{-1} .$$ (38)

B86R

X $\alpha\beta\gamma$ Re-optimised. Re-optimised $\beta$ of B86 used in part 3 of Becke's 1997 paper.
doi:10.1063/1.475007

DIRAC

Slater-Dirac Exchange Energy. Automatically generated Slater-Dirac exchange.
doi:10.1103/PhysRev.81.385

BRUEG

Becke-Roussel Exchange Functional -- Uniform Electron Gas Limit. A. D. Becke and M. R. Roussel,Phys. Rev. A 39, 3761 (1989)

As for BR but with ${\gamma=0.8}$.

VWN5

Vosko-Wilk-Nusair (1980) V local correlation energy. VWN 1980(V) functional. The fitting parameters for $\Delta\varepsilon_{c}(r_{s},\zeta)_{V}$ appear in the caption of table 7 in the reference.
doi:VWN80

VSXC

.
doi:10.1063/1.476577

B88

Becke 1988 Exchange Functional
doi:10.1103/PhysRevA.38.3098

G96

Gill's 1996 Gradient Corrected Exchange Functional
doi:G96

THGFCO

.. Density and gradient dependent first row exchange-correlation functional.
doi:10.1016/S0009-2614(97)00586-1

CS1

Colle-Salvetti correlation functional. R. Colle and O. Salvetti, Theor. Chim. Acta 37, 329 (1974); C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37, 785(1988)

CS1 is formally identical to CS2, except for a reformulation in which the terms involving $\upsilon$ are eliminated by integration by parts. This makes the functional more economical to evaluate. In the limit of exact quadrature, CS1 and CS2 are identical, but small numerical differences appear with finite integration grids.

THGFL

.. Density dependent first row exchange-correlation functional for closed shell systems.
doi:10.1016/S0009-2614(97)00586-1

HCTH93

Handy least squares fitted functional
doi:10.1063/1.477267

B97DF

Density functional part of B97. This functional needs to be mixed with 0.1943*exact exchange.
doi:10.1063/1.475007

VWN3

Vosko-Wilk-Nusair (1980) III local correlation energy. VWN 1980(III) functional
doi:VWN80

MK00B

Exchange Functional for Accurate Virtual Orbital Energies. MK00 with gradient correction of the form of B88X but with different empirical parameter.
doi:10.1063/1.481298

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