Density Functional Theory (DFT)¶

Overview¶

The NWChem density functional theory (DFT) module uses the Gaussian basis set approach to compute closed shell and open shell densities and Kohn-Sham orbitals in the:

local density approximation (LDA),
non-local density approximation (NLDA),
local spin-density approximation (LSD),
non-local spin-density approximation (NLSD),
non-local meta-GGA approximation (metaGGA),
any empirical mixture of local and non-local approximations (including exact exchange), and
asymptotically corrected exchange-correlation potentials.
spin-orbit effects

The formal scaling of the DFT computation can be reduced by choosing to use auxiliary Gaussian basis sets to fit the charge density (CD) and/or fit the exchange-correlation (XC) potential.

DFT input is provided using the compound DFT directive

 DFT  
   ...  
 END

The actual DFT calculation will be performed when the input module encounters the TASK directive.

 TASK DFT

Once a user has specified a geometry and a Kohn-Sham orbital basis set the DFT module can be invoked with no input directives (defaults invoked throughout). There are sub-directives which allow for customized application; those currently provided as options for the DFT module are:

VECTORS [[input] (<string input_movecs default atomic>) || \  
                  (project <string basisname> <string filename>)] \ 
          [swap [alpha||beta] <integer vec1 vec2> ...] \ 
          [output <string output_filename default input_movecs>] \  
 XC [[acm] [b3lyp] [beckehandh] [pbe0]\  
    [becke97]  [becke97-1] [becke97-2] [becke97-3] [becke97-d] [becke98] \  
     [hcth] [hcth120] [hcth147] [hcth147@tz2p]\
     [hcth407] [becke97gga1]  [hcth407p]\  
     [mpw91] [mpw1k] [xft97] [cft97] [ft97] [op] [bop] [pbeop]\  
     [xpkzb99] [cpkzb99] [xtpss03] [ctpss03] [xctpssh]\ 
     [b1b95] [bb1k] [mpw1b95] [mpwb1k] [pw6b95] [pwb6k] [m05] [m05-2x] [vs98] \ 
     [m06] [m06-hf] [m06-L] [m06-2x] \  
     [HFexch <real prefactor default 1.0>] \  
     [becke88 [nonlocal] <real prefactor default 1.0>] \  
     [xperdew91 [nonlocal] <real prefactor default 1.0>] \  
     [xpbe96 [nonlocal] <real prefactor default 1.0>] \  
     [gill96 [nonlocal] <real prefactor default 1.0>] \  
     [lyp <real prefactor default 1.0>] \  
     [perdew81 <real prefactor default 1.0>] \  
     [perdew86 [nonlocal] <real prefactor default 1.0>] \  
     [perdew91 [nonlocal] <real prefactor default 1.0>] \ 
     [cpbe96 [nonlocal] <real prefactor default 1.0>] \  
     [pw91lda <real prefactor default 1.0>] \ 
     [slater <real prefactor default 1.0>] \  
     [vwn_1 <real prefactor default 1.0>] \  
     [vwn_2 <real prefactor default 1.0>] \  
     [vwn_3 <real prefactor default 1.0>] \ 
     [vwn_4 <real prefactor default 1.0>] \  
     [vwn_5 <real prefactor default 1.0>] \  
     [vwn_1_rpa <real prefactor default 1.0>] \  
     [xtpss03 [nonlocal] <real prefactor default 1.0>] \ 
     [ctpss03 [nonlocal] <real prefactor default 1.0>] \  
     [bc95 [nonlocal] <real prefactor default 1.0>] \  
     [xpw6b95 [nonlocal] <real prefactor default 1.0>] \  
     [xpwb6k [nonlocal] <real prefactor default 1.0>] \  
     [xm05 [nonlocal] <real prefactor default 1.0>] \  
     [xm05-2x [nonlocal] <real prefactor default 1.0>] \  
     [cpw6b95 [nonlocal] <real prefactor default 1.0>] \  
     [cpwb6k [nonlocal] <real prefactor default 1.0>] \  
     [cm05 [nonlocal] <real prefactor default 1.0>] \  
     [cm05-2x [nonlocal] <real prefactor default 1.0>]] \  
     [xvs98 [nonlocal] <real prefactor default 1.0>]] \  
     [cvs98 [nonlocal] <real prefactor default 1.0>]] \  
     [xm06-L [nonlocal] <real prefactor default 1.0>]] \ 
     [xm06-hf [nonlocal] <real prefactor default 1.0>]] \  
     [xm06 [nonlocal] <real prefactor default 1.0>]] \ 
     [xm06-2x [nonlocal] <real prefactor default 1.0>]] \  
     [cm06-L [nonlocal] <real prefactor default 1.0>]] \  
     [cm06-hf [nonlocal] <real prefactor default 1.0>]] \  
     [cm06 [nonlocal] <real prefactor default 1.0>]] \  
     [cm06-2x [nonlocal] <real prefactor default 1.0>]]   
 CONVERGENCE [[energy <real energy default 1e-7>] \  
              [density <real density default 1e-5>] \  
              [gradient <real gradient default 5e-4>] \  
              [dampon <real dampon default 0.0>] \  
              [dampoff <real dampoff default 0.0>] \  
              [diison <real diison default 0.0>] \  
              [diisoff <real diisoff default 0.0>] \  
              [levlon <real levlon default 0.0>] \  
              [levloff <real levloff default 0.0>] \  
              [ncydp <integer ncydp default 2>] \  
              [ncyds <integer ncyds default 30>] \  
              [ncysh <integer ncysh default 30>] \  
              [damp <integer ndamp default 0>] [nodamping] \  
              [diis [nfock <integer nfock default 10>]] \  
              [nodiis] [lshift <real lshift default 0.5>] \  
              [nolevelshifting] \  
              [hl_tol <real hl_tol default 0.1>] \  
              [rabuck [n_rabuck <integer n_rabuck default 25>]\
              [fast] ]  
 GRID [(xcoarse||coarse||medium||fine||xfine||huge) default medium] \  
      [(gausleg||lebedev ) default lebedev ] \  
      [(becke||erf1||erf2||ssf) default erf1] \  
      [(euler||mura||treutler) default mura] \ 
      [rm <real rm default 2.0>] \  
      [nodisk] 
 TOLERANCES [[tight] [tol_rho <real tol_rho default 1e-10>] \  
             [accCoul <integer accCoul default 8>] \  
             [radius <real radius default 25.0>]]  
 [(LB94||CS00 <real shift default none>)]  
 DECOMP 
 ODFT 
 DIRECT  
 SEMIDIRECT [filesize <integer filesize default disksize>] 
            [memsize  <integer memsize default available>]  
            [filename <string filename default $file_prefix.aoints$>]
 INCORE 
 ITERATIONS <integer iterations default 30> 
 MAX_OVL  
 CGMIN  
 RODFT  
 MULLIKEN  
 DISP  
 XDM [ a1 <real a1> ]   [ a2 <real a2> ]  
 MULT <integer mult default 1>  
 NOIO  
 PRINT||NOPRINT
 SYM <string (ON||OFF) default ON>
 ADAPT <string (ON||OFF) default ON>

The following sections describe these keywords and optional sub-directives that can be specified for a DFT calculation in NWChem.

Specification of Basis Sets for the DFT Module¶

The DFT module requires at a minimum the basis set for the Kohn-Sham molecular orbitals. This basis set must be in the default basis set named “ao basis”, or it must be assigned to this default name using the SET directive.

In addition to the basis set for the Kohn-Sham orbitals, the charge density fitting basis set can also be specified in the input directives for the DFT module. This basis set is used for the evaluation of the Coulomb potential in the Dunlap scheme¹². The charge density fitting basis set must have the name cd basis. This can be the actual name of a basis set, or a basis set can be assigned this name using the SET directive. If this basis set is not defined by input, the O(N⁴) exact Coulomb contribution is computed.

The user also has the option of specifying a third basis set for the evaluation of the exchange-correlation potential. This basis set must have the name xc basis. If this basis set is not specified by input, the exchange contribution (XC) is evaluated by numerical quadrature. In most applications, this approach is efficient enough, so the “xc basis” basis set is not required.

For the DFT module, the input options for defining the basis sets in a given calculation can be summarized as follows:

ao basis - Kohn-Sham molecular orbitals; required for all calculations
cd basis - charge density fitting basis set; optional, but recommended for evaluation of the Coulomb potential
“xc basis” - exchange-correlation (XC) fitting basis set; optional, and not recommended

ADFT New in NWChem 7.2.0:¶

Use of the auxiliary density functional theory method (ADFT)³ can be triggered by means of the adft keyword. This can result in a large speed-up when using “pure” GGA functionals (e.g. PBE96) and Laplacian-dependent mGGA functionals (e.g. SCAN-L). The speed-up comes from the use of the fitted density obtained with the charge density fitting technique to approximate both the Coulomb and Exchange-Correlation contributions.

The ADFT method is similar in spirit to the exchange-correlation fitting technique triggered by specifying an xc basis without the adft keyword. It is important to note that, different to straight exchange-correlation fitting, energy derivatives are well-defined within the ADFT framework. As a consequence, geometry optimizations and harmonic vibrational frequencies are well-behaved.

The ADFT method requires a charge density fitting basis set (see DFT basis set section). If not cd basis set is provided, the weigend coulomb fitting basis set will be loaded.

VECTORS and MAX_OVL: KS-MO Vectors¶

The VECTORS directive is the same as that in the SCF module. Currently, the LOCK keyword is not supported by the DFT module, however the directive

 MAX_OVL

has the same effect.

XC and DECOMP: Exchange-Correlation Potentials¶

 XC [[acm] [b3lyp] [beckehandh] [pbe0] [bhlyp]\  
    [becke97]  [becke97-1] [becke97-2] [becke97-3] [becke98] [hcth] [hcth120] [hcth147] [hcth147@tz2p] \
     [hcth407] [becke97gga1] [hcth407p] \  
     [optx] [hcthp14] [mpw91] [mpw1k] [xft97] [cft97] [ft97] [op] [bop] [pbeop]\  
     [m05] [m05-2x] [m06] [m06-l] [m06-2x] [m06-hf] [m08-hx] [m08-so] [m11] [m11-l]\  
     [HFexch <real prefactor default 1.0>] \  
     [becke88 [nonlocal] <real prefactor default 1.0>] \  
     [xperdew91 [nonlocal] <real prefactor default 1.0>] \  
     [xpbe96 [nonlocal] <real prefactor default 1.0>] \  
     [gill96 [nonlocal] <real prefactor default 1.0>] \  
     [lyp <real prefactor default 1.0>] \  
     [perdew81 <real prefactor default 1.0>] \  
     [perdew86 [nonlocal] <real prefactor default 1.0>] \  
     [perdew91 [nonlocal] <real prefactor default 1.0>] \  
     [cpbe96 [nonlocal] <real prefactor default 1.0>] \  
     [pw91lda <real prefactor default 1.0>] \  
     [slater <real prefactor default 1.0>] \  
     [vwn_1 <real prefactor default 1.0>] \  
     [vwn_2 <real prefactor default 1.0>] \  
     [vwn_3 <real prefactor default 1.0>] \  
     [vwn_4 <real prefactor default 1.0>] \  
     [vwn_5 <real prefactor default 1.0>] \  
     [vwn_1_rpa <real prefactor default 1.0>]]

The user has the option of specifying the exchange-correlation treatment in the DFT Module (see table below for full list of functionals). The default exchange-correlation functional is defined as the local density approximation (LDA) for closed shell systems and its counterpart the local spin-density (LSD) approximation for open shell systems. Within this approximation, the exchange functional is the Slater ρ^1/3 functional⁴⁵, and the correlation functional is the Vosko-Wilk-Nusair (VWN) functional (functional V)⁶. The parameters used in this formula are obtained by fitting to the Ceperley and Alder Quantum Monte-Carlo solution of the homogeneous electron gas.

These defaults can be invoked explicitly by specifying the following keywords within the DFT module input directive, XC slater vwn_5.

That is, this statement in the input file

dft  
 XC slater vwn_5  
end  
task dft

is equivalent to the simple line

task dft

The DECOMP directive causes the components of the energy corresponding to each functional to be printed, rather than just the total exchange-correlation energy that is the default. You can see an example of this directive in the sample input.

Many alternative exchange and correlation functionals are available to the user as listed in the table below. The following sections describe how to use these options.

Libxc interface New in NWChem 7.2.0:¶

If NWChem is compiled by linking it with the libxc DFT library (as described in the Interfaces with External Software section), the user will be able to use most of the XC functionals available in libxc.
The input syntax requires to use the xc keyword followed by the functionals name from list available in Libxc

For example, the following input for the NWChem libxc interface

dft
 xc gga_x_pbe 1.0 gga_x_pbe 1.0
end

while trigger use of the same PBE96 functionals as in the NWChem built-in interface

dft
 xc xpbe96 1.0 cpbe96 1.0
end

Exchange-Correlation Functionals¶

There are several Exchange and Correlation functionals in addition to the default slater and vwn_5 functionals. These are either local or gradient-corrected functionals (GCA); a full list can be found in the table below.

The Hartree-Fock exact exchange functional, (which has O(N⁴) computation expense), is invoked by specifying

  XC HFexch

Note that the user also has the ability to include only the local or nonlocal contributions of a given functional. In addition, the user can specify a multiplicative prefactor (the variable in the input) for the local/nonlocal component or total. An example of this might be,

  XC becke88 nonlocal 0.72

The user should be aware that the Becke88 local component is simply the Slater exchange and should be input as such.

Any combination of the supported exchange functional options can be used. For example, the popular Gaussian B3 exchange could be specified as:

  XC slater 0.8 becke88 nonlocal 0.72 HFexch 0.2

Any combination of the supported correlation functional options can be used. For example, B3LYP could be specified as:

XC vwn_1_rpa 0.19 lyp 0.81 HFexch 0.20  slater 0.80 becke88 nonlocal 0.72

and X3LYP as:

xc vwn_1_rpa 0.129 lyp 0.871 hfexch 0.218 slater 0.782 \
becke88 nonlocal 0.542  xperdew91 nonlocal 0.167

Setting up common exchange-correlation functionals¶

B3LYP: xc b3lyp
PBE0: xc pbe0
PBE96: xc xpbe96 cpbe96
PW91: xc xperdew91 perdew91
BHLYP: xc bhlyp
Becke Half and Half: xc beckehandh
BP86: xc becke88 perdew86
BP91: xc becke88 perdew91
BLYP: xc becke88 lyp

Minnesota Functionals

xc m05
xc m05-2x
xc m06
xc m06-l
xc m06-2x
xc m06-hf
xc m08-hx
xc m08-so
xc m11
xc m11-l

Analytic second derivatives are not supported with the Minnesota functionals yet.

Combined Exchange and Correlation Functionals¶

In addition to the options listed above for the exchange and correlation functionals, the user has the alternative of specifying combined exchange and correlation functionals.

The available hybrid functionals (where a Hartree-Fock Exchange component is present) consist of the Becke “half and half”⁷, the adiabatic connection method⁸, Becke 1997 (“Becke V” paper⁹).

The keyword beckehandh specifies that the exchange-correlation energy will be computed as

E_XC ≈ ½E_X^HF + ½E_X^Slater + ½E_C^PW91LDA

We know this is NOT the correct Becke prescribed implementation that requires the XC potential in the energy expression. But this is what is currently implemented as an approximation to it.

The keyword acm specifies that the exchange-correlation energy is computed as

E_XC = a₀E_X^HF + (1 - a₀)E_X^Slater + a_XδE_X^Becke88 + E_C^VWN + a_CδE_C^Perdew91

where

a₀ = 0.20, a_X = 0.72, a_C = 0.81

and δ stands for a non-local component.

The keyword b3lyp specifies that the exchange-correlation energy is computed as

E_XC = a₀E_X^HF + (1 - a₀)E_X^Slater + a_XδE_X^Becke88 + (1 - a_C)E_C^VWN_1_RPA + a_CδE_C^LYP

where

a₀ = 0.20, a_X = 0.72, a_C = 0.81

XC Functionals Summary¶

Table 1: Available Exchange (X) and Correlation (C) functionals. GGA is the Generalized Gradient Approximation, and Meta refers to Meta-GGAs. The column 2nd refers to second derivatives of the energy with respect to nuclear position.
Keyword	X	C	GGA	Meta	Hybr.	2nd	Ref.
slater	*					Y	⁴⁵
vwn_1		*				Y	⁶
vwn_2		*				Y	⁶
vwn_3		*				Y	⁶
vwn_4		*				Y	⁶
vwn_5		*				Y	⁶
vwn_1_rpa		*				Y	⁶
perdew81		*				Y	¹⁰
pw91lda		*				Y	¹¹
xbecke86b	*		*			N	¹²
becke88	*		*			Y	¹³
xperdew86	*		*			N	¹⁴
xperdew91	*		*			Y	¹¹
xpbe96	*		*			Y	¹⁵¹⁶
gill96	*		*			Y	¹⁷
optx	*		*			N	¹⁸
mpw91	*		*			Y	¹⁹²⁰
xft97	*		*			N	²¹²²
rpbe	*		*			Y	²³
revpbe	*		*			Y	²⁴
xpw6b95	*		*			N	²⁵
xpwb6k	*		*			N	²⁵
perdew86		*	*			Y	¹⁴
lyp		*	*			Y	²⁶
perdew91		*	*			Y	²⁷²⁸
cpbe96		*	*			Y	¹⁵¹⁶
cft97		*	*			N	²¹²²
op		*	*			N	²⁹
hcth	*	*	*			N	³⁰
hcth120	*	*	*			N	³¹
hcth147	*	*	*			N	³¹
hcth147@tz2p	*	*	*			N	³²
hcth407	*	*	*			N	³³
becke97gga1	*	*	*			N	³⁴
hcthp14	*	*	*			N	³⁵
ft97	*	*	*			N	²¹²²
htch407p	*	*	*			N	³⁶
bop	*	*	*			N	²⁹
pbeop	*	*	*			N	³⁷
xpkzb99	*			*		N	³⁸
cpkzb99		*		*		N	³⁸
xtpss03	*			*		N	³⁹
ctpss03		*		*		N	³⁹
bc95		*		*		N	²³
cpw6b95		*		*		N	²⁵
cpwb6k		*		*		N	²⁵
xm05	*			*	*	N	⁴⁰⁴¹
cm05		*		*		N	⁴⁰⁴¹
m05-2x	*	*		*	*	N	⁴²
xm05-2x	*			*	*	N	⁴²
cm05-2x		*		*		N	⁴²
xctpssh				*	*	N	⁴³
bb1k				*	*	N	²⁴
mpw1b95				*	*	N	⁴⁴
mpwb1k				*	*	N	⁴⁴
pw6b95				*	*	N	²⁵
pwb6k				*	*	N	²⁵
m05				*	*	N	⁴⁰
vs98				*	*	N	⁴⁵
xvs98	*			*		N	⁴⁵
cvs98		*		*		N	⁴⁵
m06-L	*	*		*		N	⁴⁶
xm06-L	*			*		N	⁴⁶
cm06-L		*		*		N	⁴⁶
m06-hf				*	*	N	⁴⁷
xm06-hf	*			*	*	N	⁴⁷
cm06-hf		*		*		N	⁴⁷
m06				*	*	N	⁴⁸
xm06	*			*	*	N	⁴⁸
cm06		*		*		N	⁴⁸
m06-2x				*	*	N	⁴⁶
xm06-2x	*			*	*	N	⁴⁶
cm06-2x		*		*		N	⁴⁶
cm08-hx		*		*		N	⁴⁹
xm08-hx	*			*		N	⁴⁹
m08-hx	*	*		*	*	N	⁴⁹
cm08-so		*		*		N	⁴⁹
xm08-so	*			*		N	⁴⁹
m08-so	*	*		*	*	N	⁴⁹
cm11		*		*		N	⁵⁰
xm11	*			*		N	⁵⁰
m11	*	*		*	*	N	⁵⁰
cm11-l		*		*		N	⁵¹
xm11-l	*			*		N	⁵¹
m11-l	*	*		*		N	⁵¹
csogga		*	*			N	⁴⁹
xsogga	*		*			N	⁴⁹
sogga	*	*	*			N	⁴⁹
csogga11		*	*			N	⁵²
xsogga11	*		*			N	⁵²
sogga11	*	*	*			N	⁵²
csogga11-x		*				N	⁵³
xsogga11-x	*		*			N	⁵³
sogga11-x	*	*	*		*	N	⁵³
dldf	*	*		*	*	N	⁵⁴
beckehandh	*	*			*	Y	⁷
b3lyp	*	*	*		*	Y	⁸
acm	*	*	*		*	Y	⁸
becke97	*	*	*		*	N	⁹
becke97-1	*	*	*		*	N	³⁰
becke97-2	*	*	*		*	N	⁵⁵
becke97-3	*	*	*		*	N	⁵⁶
becke97-d	*	*	*		*	N	⁵⁷
becke98	*	*	*		*	N	⁵⁸
pbe0	*	*	*		*	Y	⁵⁹
mpw1k	*	*	*		*	Y	⁶⁰
xmvs15	*			*		N	⁶¹
hle16	*	*	*		*	Y	⁶²
scan	*	*	*	*		N	⁶³
scanl	*	*	*	*		N	⁶⁴
revm06-L	*	*	*	*		N	⁶⁵
revm06	*	*	*	*	*	N	⁶⁶
wb97x	*	*	*		*	N	⁶⁷
wb97x-d3	*	*	*		*	N	⁶⁸
rscan	*	*	*	*		N	⁶⁹
r2scan	*	*	*	*		N	⁷⁰
r2scan0	*	*	*	*	*	N	⁷¹
r2scanl	*	*	*	*		N	⁷²⁷³
ncap	*	*	*			Y	⁷⁴

Meta-GGA Functionals¶

One way to calculate meta-GGA energies is to use orbitals and densities from fully self-consistent GGA or LDA calculations and run them in one iteration in the meta-GGA functional. It is expected that meta-GGA energies obtained this way will be close to fully self consistent meta-GGA calculations.

It is possible to calculate metaGGA energies both ways in NWChem, that is, self-consistently or with GGA/LDA orbitals and densities. However, since second derivatives are not available for metaGGAs, in order to calculate frequencies, one must use task dft freq numerical. A sample file with this is shown below, in Sample input file. In this instance, the energy is calculated self-consistently and geometry is optimized using the analytical gradients.

(For more information on metaGGAs, see Kurth et al 1999 ⁷⁵ for a brief description of meta-GGAs, and citations 14-27 therein for thorough background)

Note: both TPSS and PKZB correlation require the PBE GGA CORRELATION (which is itself dependent on an LDA). The decision has been made to use these functionals with the accompanying local PW91LDA. The user cannot set the local part of these metaGGA functionals.

Range-Separated Functionals¶

Using the Ewald decomposition

\[\frac{1}{r_{12}} = \frac{\alpha + \beta \texttt{erf}(\mu r_{12})}{r_{12}} + \frac{1 - [\alpha + \beta \texttt{erf}(\mu r_{12})]}{r_{12}}\]

we can split the the Exchange interaction as

\[E_{X} = E_X^{LR} + E_X^{SR}\]

Therefore, the long-range HF Exchange energy becomes

\[E_X^{LR} = \alpha E_X^{HF} - \frac{\beta}{2} \sum_i \sum_j \int \int \phi_i(r_1)\phi_j(r_1)\frac{\texttt{erf}(\mu r_{12})}{r_{12}} \phi_i(r_2)\phi_j(r_2)\]

 cam <real cam> cam_alpha <real cam_alpha> cam_beta <cam_beta>

cam represents the attenuation parameter μ, cam_alpha and cam_beta are the α and β parameters that control the amount of short-range DFT and long-range HF Exchange according to the Ewald decomposition. As r₁₂ → 0, the HF exchange fraction is α, while the DFT exchange fraction is 1 - α. As r₁₂ → ∞, the HF exchange fraction approaches α + β and the DFT exchange fraction approaches 1 - α - β. In the HSE functional, the HF part is short-ranged and DFT is long-ranged.

Range separated functionals (or long-range corrected or LC) can be specified as follows:

CAM-B3LYP:

xc xcamb88 1.00 lyp 0.81 vwn_5 0.19 hfexch 1.00  
cam 0.33 cam_alpha 0.19 cam_beta 0.46

LC-BLYP:

xc xcamb88 1.00 lyp 1.0 hfexch 1.00  
cam 0.33 cam_alpha 0.0 cam_beta 1.0

LC-PBE:

xc xcampbe96 1.0 cpbe96 1.0 HFexch 1.0 
cam 0.30 cam_alpha 0.0 cam_beta 1.0

LC-PBE0 or CAM-PBE0:

xc xcampbe96 1.0 cpbe96 1.0 HFexch 1.0
cam 0.30 cam_alpha 0.25 cam_beta 0.75

BNL (Baer, Neuhauser, Lifshifts):

xc xbnl07 0.90 lyp 1.00 hfexch 1.00  
cam 0.33 cam_alpha 0.0 cam_beta 1.0

LC-wPBE:

xc xwpbe 1.00 cpbe96 1.0 hfexch 1.00  
cam 0.4 cam_alpha 0.00 cam_beta 1.00

LRC-wPBEh:

xc xwpbe 0.80 cpbe96 1.0 hfexch 1.00 
cam 0.2 cam_alpha 0.20 cam_beta 0.80

QTP-00

xc xcamb88 1.00 lyp 0.80 vwn_5 0.2 hfexch 1.00 
cam 0.29 cam_alpha 0.54 cam_beta 0.37

rCAM-B3LYP

xc xcamb88 1.00 lyp 1.0 vwn_5 0. hfexch 1.00 becke88 nonlocal 0.13590
cam 0.33 cam_alpha 0.18352 cam_beta 0.94979

HSE03 functional: 0.25*Ex(HF-SR) - 0.25*Ex(PBE-SR) + Ex(PBE) + Ec(PBE), where gamma(HF-SR) = gamma(PBE-SR)

xc hse03

or it can be explicitly set as

xc xpbe96 1.0 xcampbe96 -0.25 cpbe96 1.0 srhfexch 0.25
cam 0.33 cam_alpha 0.0 cam_beta 1.0

HSE06 functional:

xc xpbe96 1.0 xcampbe96 -0.25 cpbe96 1.0 srhfexch 0.25
cam 0.11 cam_alpha 0.0 cam_beta 1.0

Please see references ⁷⁶⁷⁷⁷⁸⁷⁹⁸⁰⁸¹⁸²⁸³⁸⁴⁸⁵⁸⁶⁸⁷⁸⁸ (not a complete list) for further details about the theory behind these functionals and applications.

Example illustrating the CAM-B3LYP functional:

start h2o-camb3lyp  
geometry units angstrom  
   O      0.00000000     0.00000000     0.11726921  
   H      0.75698224     0.00000000    -0.46907685  
   H     -0.75698224     0.00000000    -0.46907685  
end  
basis spherical  
 * library aug-cc-pvdz  
end  
dft  
 xc xcamb88 1.00 lyp 0.81 vwn_5 0.19 hfexch 1.00  
 cam 0.33 cam_alpha 0.19 cam_beta 0.46  
 direct  
 iterations 100  
end  
task dft energy

Example illustrating the HSE03 functional:

echo  
start h2o-hse  
geometry units angstrom  
O      0.00000000     0.00000000     0.11726921  
H      0.75698224     0.00000000    -0.46907685  
H     -0.75698224     0.00000000    -0.46907685  
end  
basis spherical  
 * library aug-cc-pvdz  
end  
dft  
 xc hse03  
 iterations 100  
 direct  
 end  
task dft energy

or alternatively

dft  
 xc xpbe96 1.0 xcampbe96 -0.25 cpbe96 1.0 srhfexch 0.25  
 cam 0.33 cam_alpha 0.0 cam_beta 1.0  
 iterations 100  
 direct  
end  
task dft energy

SSB-D functional¶

The SSB-D⁸⁹⁹⁰ functional is a small correction to the non-empirical PBE functional and includes a portion of Grimme’s dispersion correction (s6=0.847455). It is designed to reproduce the good results of OPBE for spin-state splittings and reaction barriers, and the good results of PBE for weak interactions. The SSB-D functional works well for these systems, including for difficult systems for DFT (dimerization of anthracene, branching of octane, water-hexamer isomers, C₁₂H₁₂ isomers, stacked adenine dimers), and for NMR chemical shieldings.

It can be specified as

xc ssb-d

Semi-empirical hybrid DFT combined with perturbative MP2¶

This theory combines hybrid density functional theory with MP2 semi-empirically. The B2PLYP functional, which is an example of this approximation, can be specified as:

mp2
 freeze atomic
end
dft  
 xc HFexch 0.53 becke88 0.47 lyp 0.73 mp2 0.27  
 dftmp2 
end

For details of the theory, please see reference⁹¹.

LB94 and CS00: Asymptotic correction¶

The keyword LB94 will correct the asymptotic region of the XC definition of exchange-correlation potential by the van-Leeuwen-Baerends exchange-correlation potential that has the correct $-1/r$ asymptotic behavior. The total energy will be computed by the XC definition of exchange-correlation functional. This scheme is known to tend to overcorrect the deficiency of most uncorrected exchange-correlation potentials.

The keyword CS00, when supplied with a real value of shift (in atomic units), will perform Casida-Salahub ‘00 asymptotic correction. This is primarily intended for use with TDDFT. The shift is normally positive (which means that the original uncorrected exchange-correlation potential must be shifted down).

When the keyword CS00 is specified without the value of shift, the program will automatically supply it according to the semi-empirical formula of Zhan, Nichols, and Dixon (again, see TDDFT for more details and references). As the Zhan’s formula is calibrated against B3LYP results, it is most meaningful to use this with the B3LYP functional, although the program does not prohibit (or even warn) the use of any other functional.

Sample input files of asymptotically corrected TDDFT calculations can be found in the corresponding section.

Sample input file¶

A simple example calculates the geometry of water, using the metaGGA functionals xtpss03 and ctpss03. This also highlights some of the print features in the DFT module. Note that you must use the line task dft freq numerical because analytic hessians are not available for the metaGGAs:

title "WATER 6-311G* meta-GGA XC geometry"  
echo  
geometry units angstroms  
 O       0.0  0.0  0.0  
 H       0.0  0.0  1.0  
 H       0.0  1.0  0.0  
end  
basis  
 H library 6-311G*  
 O library 6-311G*  
end  
dft  
 iterations 50  
 print  kinetic_energy  
 xc xtpss03 ctpss03  
 decomp  
end  
task dft optimize   
task dft freq numerical

ITERATIONS or MAXITER: Number of SCF iterations¶

 ITERATIONS or MAXITER <integer iterations default 30>

The default optimization in the DFT module is to iterate on the Kohn-Sham (SCF) equations for a specified number of iterations (default 30). The keyword that controls this optimization is ITERATIONS, and has the following general form,

  iterations <integer iterations default 30>

or

  maxiter <integer iterations default 30>

The optimization procedure will stop when the specified number of iterations is reached or convergence is met. See an example that uses this directive in Sample input file.

CONVERGENCE: SCF Convergence Control¶

 CONVERGENCE [energy <real energy default 1e-6>] \  
             [density <real density default 1e-5>] \  
             [gradient <real gradient default 5e-4>] \  
             [hl_tol <real hl_tol default 0.1>]  
             [dampon <real dampon default 0.0>] \  
             [dampoff <real dampoff default 0.0>] \  
             [ncydp <integer ncydp default 2>] \  
             [ncyds <integer ncyds default 30>] \  
             [ncysh <integer ncysh default 30>] \  
             [damp <integer ndamp default 0>] [nodamping] \  
             [diison <real diison default 0.0>] \  
             [diisoff <real diisoff default 0.0>] \  
             [(diis [nfock <integer nfock default 10>]) || nodiis] \  
             [levlon <real levlon default 0.0>] \  
             [levloff <real levloff default 0.0>] \  
             [(lshift <real lshift default 0.5>) || nolevelshifting] \  
             [rabuck [n_rabuck <integer n_rabuck default 25>] \
             [fast] ]

Convergence is satisfied by meeting any or all of three criteria;

convergence of the total energy; this is defined to be when the total DFT energy at iteration N and at iteration N-1 differ by a value less than a threshold value (the default is 1e-6). This value can be modified using the key word,

       CONVERGENCE energy <real energy default 1e-6>

convergence of the total density; this is defined to be when the total DFT density matrix at iteration N and at iteration N-1 have a RMS difference less than some value (the default is 1e-5). This value can be modified using the keyword,

       CONVERGENCE density <real density default 1e-5>

convergence of the orbital gradient; this is defined to be when the DIIS error vector becomes less than some value (the default is 5e-4). This value can be modified using the keyword,

       CONVERGENCE gradient <real gradient default 5e-4>

The default optimization strategy is to immediately begin direct inversion of the iterative subspace. In addition, if the HOMO - LUMO gap is small and the Fock matrix diagonally dominant, then level-shifting is automatically initiated. There are a variety of ways to customize this procedure to whatever is desired.

An alternative optimization strategy is to specify, by using the change in total energy (between iterations N and N-1), when to turn damping, level-shifting, and/or DIIS on/off. Start and stop keywords for each of these is available as,

 CONVERGENCE  [dampon <real dampon default 0.0>] \  
              [dampoff <real dampoff default 0.0>] \  
              [diison <real diison default 0.0>] \  
              [diisoff <real diisoff default 0.0>] \  
              [levlon <real levlon default 0.0>] \  
              [levloff <real levloff default 0.0>]

So, for example, damping, DIIS, and/or level-shifting can be turned on/off as desired.

Another strategy can be to specify how many iterations (cycles) you wish each type of procedure to be used. The necessary keywords to control the number of damping cycles (ncydp), the number of DIIS cycles (ncyds), and the number of level-shifting cycles (ncysh) are input as,

 CONVERGENCE  [ncydp <integer ncydp default 2>] \  
              [ncyds <integer ncyds default 30>] \  
              [ncysh <integer ncysh default 0>]

The amount of damping, level-shifting, time at which level-shifting is automatically imposed, and Fock matrices used in the DIIS extrapolation can be modified by the following keywords

 CONVERGENCE  [damp <integer ndamp default 0>] \  
              [diis [nfock <integer nfock default 10>]] \  
              [lshift <real lshift default 0.5>] \  
              [hl_tol <real hl_tol default 0.1>]]

CONVERGENCE DAMP Keyword¶

Damping is defined to be the percentage of the previous iterations density mixed with the current iterations density. So, for example

 CONVERGENCE damp 70

would mix 30% of the current iteration density with 70% of the previous iteration density.

CONVERGENCE LSHIFT Keyword¶

Level-Shifting is defined as the amount of shift applied to the diagonal elements of the unoccupied block of the Fock matrix. The shift is specified by the keyword lshift. For example the directive,

 CONVERGENCE lshift 0.5

causes the diagonal elements of the Fock matrix corresponding to the virtual orbitals to be shifted by 0.5 a.u. By default, this level-shifting procedure is switched on whenever the HOMO-LUMO gap is small. Small is defined by default to be 0.05 au but can be modified by the directive hl_tol. An example of changing the HOMO-LUMO gap tolerance to 0.01 would be,

 CONVERGENCE hl_tol 0.01

CONVERGENCE DIIS Keyword¶

Direct inversion of the iterative subspace with extrapolation of up to 10 Fock matrices is a default optimization procedure. For large molecular systems the amount of available memory may preclude the ability to store this number of N² arrays in global memory. The user may then specify the number of Fock matrices to be used in the extrapolation (must be greater than three (3) to be effective). To set the number of Fock matrices stored and used in the extrapolation procedure to 3 would take the form,

 CONVERGENCE diis 3

The user has the ability to simply turn off any optimization procedures deemed undesirable with the obvious keywords,

 CONVERGENCE [nodamping] [nodiis] [nolevelshifting]

For systems where the initial guess is very poor, the user can try using fractional occupation of the orbital levels during the initial cycles of the SCF convergence⁹². The input has the following form

 CONVERGENCE rabuck [n_rabuck <integer n_rabuck default 25>]]

where the optional value n_rabuck determines the number of SCF cycles during which the method will be active. For example, to set equal to 30 the number of cycles where the Rabuck method is active, you need to use the following line

 CONVERGENCE rabuck 30

CONVERGENCE FAST Keyword¶

The convergence fast option turns on a series of parameters that most often speed-up convergence, but not in 100% of the cases.

  CONVERGENCE fast

Here is an input snippet that would give you the same result as convergence fast

dft
  convergence lshift 0. ncydp 0 dampon 1d99 dampoff 1d-4 damp 40
end
set quickguess t
task dft

CDFT: Constrained DFT¶

This option enables the constrained DFT formalism by Wu and Van Voorhis⁹³:

 CDFT <integer fatom1 latom1> [<integer fatom2 latom2>] (charge||spin <real constaint_value>) \ 
      [pop (becke||mulliken||lowdin) default lowdin]

Variables fatom1 and latom1 define the first and last atom of the group of atoms to which the constraint will be applied. Therefore, the atoms in the same group should be placed continuously in the geometry input. If fatom2 and latom2 are specified, the difference between group 1 and 2 (i.e. 1-2) is constrained.

The constraint can be either on the charge or the spin density (number of alpha - beta electrons) with a user specified constraint_value. Note: No gradients have been implemented for the spin constraints case. Geometry optimizations can only be performed using the charge constraint.

To calculate the charge or spin density, the Becke, Mulliken, and Lowdin population schemes can be used. The Lowdin scheme is default while the Mulliken scheme is not recommended. If basis sets with many diffuse functions are used, the Becke population scheme is recommended.

Multiple constraints can be defined simultaniously by defining multiple cdft lines in the input. The same population scheme will be used for all constraints and only needs to be specified once. If multiple population options are defined, the last one will be used. When there are convergence problems with multiple constraints, the user is advised to do one constraint first and to use the resulting orbitals for the next step of the constrained calculations.

It is best to put convergence nolevelshifting in the dft directive to avoid issues with gradient calculations and convergence in CDFT. Use orbital swap to get a broken-symmetry solution.

An input example is given below.

geometry 
symmetry 
 C  0.0  0.0  0.0  
 O  1.2  0.0  0.0  
 C  0.0  0.0  2.0  
 O  1.2  0.0  2.0  
end  
basis  
 * library 6-31G* 
end  
dft  
 xc b3lyp  
 convergence nolevelshifting  
 odft  
 mult 1  
 vectors swap beta 14 15  
 cdft 1 2 charge 1.0  
end  
task dft

SMEAR: Fractional Occupation of the Molecular Orbitals¶

The SMEAR keyword is useful in cases with many degenerate states near the HOMO (eg metallic clusters)

 SMEAR <real smear default 0.001>

This option allows fractional occupation of the molecular orbitals. A Gaussian broadening function of exponent smear is used as described in the paper by Warren and Dunlap⁹⁴. The user must be aware that an additional energy term is added to the total energy in order to have energies and gradients consistent.

FON: Calculations with fractional numbers of electrons¶

Restricted¶

fon partial 3 electrons 1.8 filled 2

Here 1.8 electrons will be equally divided over 3 valence orbitals and 2 orbitals are fully filled. The total number of electrons here is 5.8

Example input:

echo 
title "carbon atom" 
start carbon_fon 
geometry  
symmetry c1  
  C   0.0 0.0 0.0  
end  
basis  
 * library 6-31G  
end
dft 
 direct  
 grid xfine  
 convergence energy 1d-8  
 xc pbe0  
 fon partial 3 electrons 1.8 filled 2 
end
task dft energy

Unrestricted¶

fon alpha partial 3 electrons 0.9 filled 2 
fon beta partial 3 electrons 0.9 filled 2

Here 0.9 electrons will be equally divided over 3 alpha valence orbitals and 2 alpha orbitals are fully filled. Similarly for beta. The total number of electrons here is 5.8

Example input:

echo  
title "carbon atom"  
start carbon_fon 
geometry  
 C   0.0 0.0 0.0  
end  
basis  
 * library 6-31G  
end  
dft 
 odft  
 fon alpha partial 3 electrons 0.9 filled 2  
 fon beta partial 3 electrons 0.9 filled 2  
end  
task dft energy

To set fractional numbers in the core orbitals, add the following directive in the input file:

   set dft:core_fon .true.

Example input:

dft
 print "final vectors analysis"
 odft
 direct
 fon alpha partial 2 electrons 1.0 filled 2
 fon beta partial 2 electrons 1.0 filled 2
 xc pbe0
 convergence energy 1d-8
end
task dft

OCCUP: Controlling the occupations of molecular orbitals¶

Example:

echo  
start h2o_core_hole  
memory 1000 mb  
geometry units au  
 O 0       0        0  
 H 0       1.430   -1.107  
 H 0      -1.430   -1.107  
end  
basis  
 O library 6-31g*  
 H library 6-31g*  
end  
occup     
 6 6      # occupation list for 6 alpha and 6 beta orbitals 
 1.0 0.0  # core-hole in the first beta orbital
 1.0 1.0  
 1.0 1.0  
 1.0 1.0  
 1.0 1.0  
 0.0 0.0  
end  
dft  
 odft  
 mult 1  
 xc beckehandh  
end  
task dft

GRID: Numerical Integration of the XC Potential¶

 GRID [(xcoarse||coarse||medium||fine||xfine||huge) default medium] \  
      [(gausleg||lebedev ) default lebedev ] \  
      [(becke||erf1||erf2||ssf) default erf1] \  
      [(euler||mura||treutler) default mura] \  
      [rm <real rm default 2.0>] \  
      [nodisk]

A numerical integration is necessary for the evaluation of the exchange-correlation contribution to the density functional. The default quadrature used for the numerical integration is an Euler-MacLaurin scheme for the radial components (with a modified Mura-Knowles transformation) and a Lebedev scheme for the angular components. Within this numerical integration procedure various levels of accuracy have been defined and are available to the user. The user can specify the level of accuracy with the keywords; xcoarse, coarse, medium, fine, xfine and huge. The default is medium.

 GRID [xcoarse||coarse||medium||fine||xfine||huge]

Our intent is to have a numerical integration scheme which would give us approximately the accuracy defined below regardless of molecular composition.

Keyword	Total Energy Target Accuracy
xcoarse	1⋅10^-4
coarse	1⋅10^-5
medium	1⋅10^-6
fine	1⋅10^-7
xfine	1⋅10^-8
huge	1⋅10^-10

In order to determine the level of radial and angular quadrature needed to give us the target accuracy, we computed total DFT energies at the LDA level of theory for many homonuclear atomic, diatomic and triatomic systems in rows 1-4 of the periodic table. In each case all bond lengths were set to twice the Bragg-Slater radius. The total DFT energy of the system was computed using the converged SCF density with atoms having radial shells ranging from 35-235 (at fixed 48/96 angular quadratures) and angular quadratures of 12/24-48/96 (at fixed 235 radial shells). The error of the numerical integration was determined by comparison to a “best” or most accurate calculation in which a grid of 235 radial points 48 theta and 96 phi angular points on each atom was used. This corresponds to approximately 1 million points per atom. The following tables were empirically determined to give the desired target accuracy for DFT total energies. These tables below show the number of radial and angular shells which the DFT module will use for for a given atom depending on the row it is in (in the periodic table) and the desired accuracy. Note, differing atom types in a given molecular system will most likely have differing associated numerical grids. The intent is to generate the desired energy accuracy (at the expense of speed of the calculation).

Keyword	Radial	Angular
xcoarse	21	194
coarse	35	302
medium	49	434
fine	70	590
xfine	100	1202

Program default number of radial and angular shells empirically determined for Row 1 atoms (Li → F) to reach the desired accuracies.

Keyword	Radial	Angular
xcoarse	42	194
coarse	70	302
medium	88	434
fine	123	770
xfine	125	1454
huge	300	1454

Program default number of radial and angular shells empirically determined for Row 2 atoms (Na → Cl) to reach the desired accuracies.

Keyword	Radial	Angular
xcoarse	75	194
coarse	95	302
medium	112	590
fine	130	974
xfine	160	1454
huge	400	1454

Program default number of radial and angular shells empirically determined for Row 3 atoms (K → Br) to reach the desired accuracies.

Keyword	Radial	Angular
xcoarse	84	194
coarse	104	302
medium	123	590
fine	141	974
xfine	205	1454
huge	400	1454

Program default number of radial and angular shells empirically determined for Row 4 atoms (Rb → I) to reach the desired accuracies.

Angular grids¶

In addition to the simple keyword specifying the desired accuracy as described above, the user has the option of specifying a custom quadrature of this type in which ALL atoms have the same grid specification. This is accomplished by using the gausleg keyword.

Gauss-Legendre angular grid¶

 GRID gausleg <integer nradpts default 50> <integer nagrid default 10>

In this type of grid, the number of phi points is twice the number of theta points. So, for example, a specification of,

 GRID gausleg 80 20

would be interpreted as 80 radial points, 20 theta points, and 40 phi points per center (or 64000 points per center before pruning).

Lebedev angular grid¶

A second quadrature is the Lebedev scheme for the angular components. Within this numerical integration procedure various levels of accuracy have also been defined and are available to the user. The input for this type of grid takes the form,

 GRID lebedev <integer radpts > <integer iangquad >

In this context the variable iangquad specifies a certain number of angular points as indicated by the table below:

Table 2: List of Lebedev quadratures
IANGQUAD	N_angular	l
1	38	9
2	50	11
3	74	13
4	86	15
5	110	17
6	146	19
7	170	21
8	194	23
9	230	25
10	266	27
11	302	29
12	350	31
13	434	35
14	590	41
15	770	47
16	974	53
17	1202	59
18	1454	65
19	1730	71
20	2030	77
21	2354	83
22	2702	89
23	3074	95
24	3470	101
25	3890	107
26	4334	113
27	4802	119
28	5294	125
29	5810	131

Therefore the user can specify any number of radial points along with the level of angular quadrature (1-29).

The user can also specify grid parameters specific for a given atom type: parameters that must be supplied are: atom tag and number of radial points. As an example, here is a grid input line for the water molecule

grid lebedev 80 11 H 70 8  O 90 11

Partitioning functions¶

GRID [(becke||erf1||erf2||ssf) default erf1]

becke : see paper⁹⁵
ssf : see paper⁹⁶
erf1 : modified ssf
erf2 : modified ssf

Erfn partitioning functions

\[\begin{array}{lcl} w_A(r) & = & \prod_{B\neq A}\frac{1}{2} \left[1 \ - \ erf(\mu^\prime_{AB})\right] \\ \mu^\prime_{AB} & = & \frac{1}{\alpha} \ \frac{\mu_{AB}}{(1-\mu_{AB}^2)^n} \\ \mu_{AB} & = & \frac{{\mathbf r}_A - {\mathbf r}_B} {\left|{\mathbf r}_A - {\mathbf r}_B \right|} \end{array}\]

Radial grids¶

 GRID [[euler||mura||treutler]  default mura]

euler : Euler-McLaurin quadrature with the transformation devised by Murray et al⁹⁷.
mura : Modification of the Murray-Handy-Laming scheme (we are not using the same scaling factors proposed in the paper by Mura and Knowles⁹⁸).
treutler : Gauss-Chebyshev using the transformation suggested by Treutler⁹⁹.

Disk usage for Grid¶

 NODISK

This keyword turns off storage of grid points and weights on disk.

TOLERANCES: Screening tolerances¶

 TOLERANCES [[tight] [tol_rho <real tol_rho default 1e-10>] \  
             [accCoul <integer accCoul default 8>] \  
             [radius <real radius default 25.0>]]

The user has the option of controlling screening for the tolerances in the integral evaluations for the DFT module. In most applications, the default values will be adequate for the calculation, but different values can be specified in the input for the DFT module using the keywords described below.

The input parameter accCoul is used to define the tolerance in Schwarz screening for the Coulomb integrals. Only integrals with estimated values greater than 10^(-accCoul) are evaluated.

 TOLERANCES accCoul <integer accCoul default 8>

Screening away needless computation of the XC functional (on the grid) due to negligible density is also possible with the use of,

 TOLERANCES tol_rho <real tol_rho default 1e-10>

XC functional computation is bypassed if the corresponding density elements are less than tol_rho.

A screening parameter, radius, used in the screening of the Becke or Delley spatial weights is also available as,

 TOLERANCES radius <real radius default 25.0>

where radius is the cutoff value in bohr.

The tolerances as discussed previously are insured at convergence. More sleazy tolerances are invoked early in the iterative process which can speed things up a bit. This can also be problematic at times because it introduces a discontinuity in the convergence process. To avoid use of initial sleazy tolerances the user can invoke the tight option:

 TOLERANCES tight

This option sets all tolerances to their default/user specified values at the very first iteration.

DIRECT, SEMIDIRECT and NOIO: Hardware Resource Control¶

 DIRECT||INCORE  
 SEMIDIRECT [filesize <integer filesize default disksize>]  
            [memsize  <integer memsize default available>]  
            [filename <string filename default $file_prefix.aoints$]  
 NOIO

The inverted charge-density and exchange-correlation matrices for a DFT calculation are normally written to disk storage. The user can prevent this by specifying the keyword noio within the input for the DFT directive. The input to exercise this option is as follows,

  noio

If this keyword is encountered, then the two matrices (inverted charge-density and exchange-correlation) are computed “on-the-fly” whenever needed.

The INCORE option is always assumed to be true but can be overridden with the option DIRECT in which case all integrals are computed “on-the-fly”.

The SEMIDIRECT option controls caching of integrals. A full description of this option is described in the Hartree-Fock section. Some functionality which is only compatible with the DIRECT option will not, at present, work when using SEMIDIRECT.

ODFT and MULT: Open shell systems¶

 ODFT
 MULT <integer mult default 1>

Both closed-shell and open-shell systems can be studied using the DFT module. Specifying the keyword MULT within the DFT directive allows the user to define the spin multiplicity of the system. The form of the input line is as follows;

  MULT <integer mult default 1>

When the keyword MULT is specified, the user can define the integer variable mult, where mult is equal to the number of alpha electrons minus beta electrons, plus 1.

When MULT is set to a negative number. For example, if MULT = -3, a triplet calculation will be performed with the beta electrons preferentially occupied. For MULT = 3, the alpha electrons will be preferentially occupied.

The keyword ODFT is unnecessary except in the context of forcing a singlet system to be computed as an open shell system (i.e., using a spin-unrestricted wavefunction).

CGMIN: Quadratic convergence algorithm¶

The cgmin keyword will use the quadratic convergence algorithm. It is possible to turn the use of the quadratic convergence algorithm off with the nocgmin keyword.

RODFT: Restricted open-shell DFT¶

The rodft keyword will perform restricted open-shell calculations. This keyword can only be used with the CGMIN keyword.

SIC: Self-Interaction Correction¶

sic [perturbative || oep || oep-loc ]
<default perturbative>

The Perdew and Zunger¹⁰ method to remove the self-interaction contained in many exchange-correlation functionals has been implemented with the Optimized Effective Potential method¹⁰⁰¹⁰¹ within the Krieger-Li-Iafrate approximation¹⁰²¹⁰³¹⁰⁴. Three variants of these methods are included in NWChem:

sic perturbative This is the default option for the sic directive. After a self-consistent calculation, the Kohn-Sham orbitals are localized with the Foster-Boys algorithm (see section on orbital localization) and the self-interaction energy is added to the total energy. All exchange-correlation functionals implemented in the NWChem can be used with this option.
sic oep With this option the optimized effective potential is built in each step of the self-consistent process. Because the electrostatic potential generated for each orbital involves a numerical integration, this method can be expensive.
sic oep-loc This option is similar to the oep option with the addition of localization of the Kohn-Sham orbitals in each step of the self-consistent process.

With oep and oep-loc options a xfine grid (see section about numerical integration ) must be used in order to avoid numerical noise, furthermore the hybrid functionals can not be used with these options. More details of the implementation of this method can be found in the paper by Garza¹⁰⁵. The components of the sic energy can be printed out using:

print "SIC information"

MULLIKEN: Mulliken analysis¶

Mulliken analysis of the charge distribution is invoked by the keyword:

 MULLIKEN

When this keyword is encountered, Mulliken analysis of both the input density as well as the output density will occur. For example, to perform a mulliken analysis and print the explicit population analysis of the basis functions, use the following

dft
 mulliken
 print "mulliken ao"
end
task dft

FUKUI: Fukui Indices¶

Fukui indices analysis is invoked by the keyword:

 FUKUI

When this keyword is encounters, the condensed Fukui indices will be calculated and printed in the output. Detailed information about the analysis can be obtained using the following

 dft  
   fukui 
   print "Fukui information"  
 end  
 task dft

The implementation of the Fukui analyis in NWChem was based on the papers by Galvan¹⁰⁶ and by Chamorro¹⁰⁷.
This implementation makes use of the generalized Fukui indices ($f_{SN}, f_{NS}, f_{SS}$).
The traditional, spin-restricted, Fukui indices are given by , and their average: $f_A^0 = \frac{1}{2}\left( f_{NN}^+ + f_{NN}^- \right)$

BSSE: Basis Set Superposition Error¶

Particular care is required to compute BSSE by the counter-poise method for the DFT module. In order to include terms deriving from the numerical grid used in the XC integration, the user must label the ghost atoms not just bq, but bq followed by the given atomic symbol. For example, the first component needed to compute the BSSE for the water dimer, should be written as follows

geometry h2o autosym units au
 O        0.00000000     0.00000000     0.22143139
 H        1.43042868     0.00000000    -0.88572555
 H       -1.43042868     0.00000000    -0.88572555
 bqH      0.71521434     0.00000000    -0.33214708
 bqH     -0.71521434     0.00000000    -0.33214708
 bqO      0.00000000     0.00000000    -0.88572555
end
basis
 H library aug-cc-pvdz
 O library aug-cc-pvdz
 bqH library H aug-cc-pvdz
 bqO library O aug-cc-pvdz
end

Please note that the ghost oxygen atom has been labeled bqO, and not just bq.

DISP: Empirical Long-range Contribution (vdW)¶

 DISP \
      [ vdw <real vdw integer default 2]] \
      [[s6 <real s6 default depends on XC functional>] \
      [ alpha <real alpha default 20.0d0] \
      [ off ]

When systems with high dependence on van der Waals interactions are computed, the dispersion term may be added empirically through long-range contribution DFT-D, i.e. E_DFT-D=E_DFT-KS+E_disp, where:

\[E_{disp}=-s_6\sum_{i=1}^{N_{atom}-1}\sum_{j=i+1}^{N_{atom}} \frac{C_{6}^{ij}}{R_{ij}^{6}} \frac{1}{ 1+e^{-\alpha \frac{R_{ij}}{R_{vdw}}}}\]

In this equation, the s₆ term depends in the functional and basis set used, C₆^ij is the dispersion coefficient between pairs of atoms. R_vdw and R_ij are related with van der Waals atom radii and the nucleus distance respectively. The α value contributes to control the corrections at intermediate distances.

There are available three ways to compute C⁶_ij:

$\(C_6^{ij}= \frac{2(C_6^{i}C_6^{j})^{2/3}(N_{eff i}N_{eff j})^{1/3}} {C_6^{i}(N_{eff i}^2)^{1/3}+(C_6^{i}N_{eff j}^2)^{1/3}}$\) where N_eff and C₆ are obtained from references ¹⁰⁸ and ¹⁰⁹ (Use vdw 0)
$\(C_6^{ij}=2\ \frac{C_6^{i}C_6^{j}}{C_6^{i}+C_6^{j}}$\) See details in reference¹¹⁰. (Use vdw 1)
$\(C_6^{ij}=\sqrt{C_6^{i}C_6^{j}}$\) See details in reference⁹¹. (Use vdw 2)

Note that in each option there is a certain set of C₆ and R_vdw. Also note that Grimme only defined parameters for elements up to Z=54 for the dispersion correction above. C₆ values for elements above Z=54 have been set to zero.

For options vdw 1 and vdw 2 , there are s₆ values by default for some functionals and triple-zeta plus double polarization basis set (TZV2P):

vdw 1 BLYP 1.40, PBE 0.70 and BP86 1.30.
vdw 2 BLYP 1.20, PBE 0.75, BP86 1.05, B3LYP 1.05, Becke97-D 1.25 and TPSS 1.00.

Grimme’s DFT-D3 is also available. Here the dispersion term has the following form:

\[ E_{disp} = \sum_{i,j} \sum_{n=6,8} s_n \ \frac{C^{ij}_n}{r_{ij}} \biggl\lbrace 1 + 6 \bigl[\frac{r_{ij}}{s_{r,n} \ R_0^{ij}} \bigr]^{-\alpha n} \biggr\rbrace^{-1}\]

This new dispersion correction covers elements through Z=94. C^ij_n (n=6,8) are coordination and geometry dependent. Details about the functional form can be found in reference ¹¹¹.

To use the Grimme DFT-D3 dispersion correction, use the option

vdw 3 (s6 and alpha cannot be set manually). Functionals for which DFT-D3 is available in NWChem are BLYP, B3LYP, BP86, Becke97-D, PBE96, TPSS, PBE0, B2PLYP, BHLYP, TPSSH, PWB6K, B1B95, SSB-D, MPW1B95, MPWB1K, M05, M05-2X, M06L, M06, M06-2X, and M06HF
vdw 4 triggers the DFT-D3BJ dispersion model. Currently only BLYP, B3LYP, BHLYP, TPSS, TPSSh, B2-PLYP, B97-D, BP86, PBE96, PW6B95, revPBE, B3PW91, pwb6k, b1b95, CAM-B3LYP, LC-wPBE, HCTH120, MPW1B95, BOP, OLYP, BPBE, OPBE and SSB are supported.

This capability is also supported for energy gradients and Hessian. Is possible to be deactivated with OFF.

NOSCF: Non Self-Consistent Calculations¶

The noscf keyword can be used to to calculate the non self-consistent energy for a set of input vectors. For example, the following input shows how a non self-consistent B3LYP energy can be calculated using a self-consistent set of vectors calculated at the Hartree-Fock level.

start h2o-noscf

geometry units angstrom
  O      0.00000000     0.00000000     0.11726921
  H      0.75698224     0.00000000    -0.46907685
  H     -0.75698224     0.00000000    -0.46907685
end

basis spherical
  * library aug-cc-pvdz
end
dft
  xc hfexch
  vectors output hf.movecs 
end
task dft energy
dft
  xc b3lyp
  vectors input hf.movecs 
  noscf 
end
task dft energy

XDM: Exchange-hole dipole moment dispersion model¶

XDM  [ a1 <real a1> ]   [ a2 <real a2> ]

See details (including list of a1 and a2 parameters) in paper ¹¹² and the website https://erin-r-johnson.github.io/software/

    geometry 
      O         -0.190010095135     -1.168397415155      0.925531922479
      H         -0.124425719598     -0.832776238160      1.818190662986
      H         -0.063897685990     -0.392575837594      0.364048725248
      O          0.174717244879      1.084630474836     -0.860510672419
      H         -0.566281023931      1.301941006866     -1.427261487135
      H          0.935093179777      1.047335209207     -1.441842151158
    end

    basis spherical
      * library aug-cc-pvdz
    end

    dft
       direct
       xc b3lyp
       xdm a1 0.6224 a2 1.7068
    end

    task dft optimize

Print Control¶

 PRINT||NOPRINT

The PRINT||NOPRINT options control the level of output in the DFT. Please see some examples using this directive in Sample input file. Known controllable print options are:

Name	Print Level	Description
“all vector symmetries”	high	symmetries of all molecular orbitals
“alpha partner info”	high	unpaired alpha orbital analysis
“common”	debug	dump of common blocks
“convergence”	default	convergence of SCF procedure
“coulomb fit”	high	fitting electronic charge density
“dft timings”	high
“final vectors”	high
“final vectors analysis”	high	print all orbital energies and orbitals
“final vector symmetries”	default	symmetries of final molecular orbitals
“information”	low	general information
“initial vectors”	high
“intermediate energy info”	high
“intermediate evals”	high	intermediate orbital energies
“intermediate fock matrix”	high
“intermediate overlap”	high	overlaps between the alpha and beta sets
“intermediate S2”	high	values of S2
“intermediate vectors”	high	intermediate molecular orbitals
“interm vector symm”	high	symmetries of intermediate orbitals
“io info”	debug	reading from and writing to disk
“kinetic_energy”	high	kinetic energy
“mulliken ao”	high	mulliken atomic orbital population
“multipole”	default	moments of alpha, beta, and nuclear charge densities
“parameters”	default	input parameters
“quadrature”	high	numerical quadrature
“schwarz”	high	integral screening info & stats at completion
“screening parameters”	high	integral accuracies
“semi-direct info”	default	semi direct algorithm

DFT Print Control Specifications

Spin-Orbit Density Functional Theory (SODFT)¶

The spin-orbit DFT module (SODFT) in the NWChem code allows for the variational treatment of the one-electron spin-orbit operator within the DFT framework. Calculations can be performed either with an electron relativistic approach (ZORA) or with an effective core potential (ECP) and a matching spin-orbit potential (SO). The current implementation does NOT use symmetry.

The actual SODFT calculation will be performed when the input module encounters the TASK directive (TASK).

 TASK SODFT

Input parameters are the same as for the DFT. Some of the DFT options are not available in the SODFT. These are max_ovl and sic.

Besides using the standard ECP and basis sets, see Effective Core Potentials for details, one also has to specify a spin-orbit (SO) potential. The input specification for the SO potential can be found in Effective Core Potentials. At this time we have not included any spin-orbit potentials in the basis set library. However, one can get these from the Stuttgart/Köln web pages http://www.tc.uni-koeln.de/PP/clickpse.en.html.

Note: One should use a combination of ECP and SO potentials that were designed for the same size core, i.e., don’t use a small core ECP potential with a large core SO potential (it will produce erroneous results).

The following is an example of a calculation of UO₂:

start uo2_sodft  
echo  

charge 2  
geometry 
 U     0.00000      0.00000     0.00000  
 O     0.00000      0.00000     1.68000  
 O     0.00000      0.00000    -1.68000  
end  
basis "ao basis"  
 * library "stuttgart rlc ecp"
END
ECP
 * library "stuttgart rlc ecp"
END
SO  
 U p  
 2    3.986181      1.816350  
 2    2.000160     11.543940  
 2    0.960841      0.794644  
 U d  
 2    4.147972      0.353683  
 2    2.234563      3.499282  
 2    0.913695      0.514635  
 U f  
 2    3.998938      4.744214  
 2    1.998840     -5.211731  
 2    0.995641      1.867860  
END  
dft  
 mult 1  
 xc hfexch  
end  
task sodft

SYM and ADAPT¶

The options SYM and ADAPT works the same way as the analogous options for the SCF code. Therefore please use the following links for SYM and ADAPT, respectively.

References¶

Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. On First-Row Diatomic Molecules and Local Density Models. The Journal of Chemical Physics 1979, 71 (12), 4993. https://doi.org/10.1063/1.438313. ↩
Eichkorn, K.; Treutler, O.; Öhm, H.; Häser, M.; Ahlrichs, R. Auxiliary Basis Sets to Approximate Coulomb Potentials. Chemical Physics Letters 1995, 240 (4), 283–290. https://doi.org/10.1016/0009-2614(95)00621-a. ↩
Köster, A. M.; Reveles, J. U.; Campo, J. M. del. Calculation of Exchange-Correlation Potentials with Auxiliary Function Densities. The Journal of Chemical Physics 2004, 121 (8), 3417–3424. https://doi.org/10.1063/1.1771638. ↩
Slater, J. C.; Johnson, K. H. Self-Consistent-Field Xα Cluster Method for Polyatomic Molecules and Solids. Phys. Rev. B 1972, 5, 844–853. https://doi.org/10.1103/PhysRevB.5.844. ↩↩
Slater, J. C. Quantum Theory of Molecules and Solids: The Self-Consistent Field for Molecules and Solids; McGraw-Hill Education, 1974; p 640. ↩↩
Vosko, S. H.; Wilk, L.; Nusair, M. Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis. Canadian Journal of physics 1980, 58 (8), 1200–1211. https://doi.org/10.1139/p80-159. ↩↩↩↩↩↩↩
Becke, A. D. A New Mixing of Hartree-Fock and Local Density-Functional Theories. The Journal of Chemical Physics 1993, 98 (2), 1372–1377. https://doi.org/https://aip.scitation.org/doi/abs/10.1063/1.464304. ↩↩
Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. The Journal of Chemical Physics 1993, 98 (7), 5648–5652. https://doi.org/10.1063/1.464913. ↩↩↩
Becke, A. D. Density-Functional Thermochemistry. V. Systematic Optimization of Exchange-Correlation Functionals. The Journal of Chemical Physics 1997, 107 (20), 8554–8560. https://doi.org/10.1063/1.475007. ↩↩
Perdew, J. P.; Zunger, A. Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems. Phys. Rev. B 1981, 23, 5048–5079. https://doi.org/10.1103/PhysRevB.23.5048. ↩↩
Perdew, J. P.; Wang, Y. Accurate and Simple Analytic Representation of the Electron-Gas Correlation Energy. Phys. Rev. B 1992, 45, 13244–13249. https://doi.org/10.1103/PhysRevB.45.13244. ↩↩
Becke, A. D. On the Large-Gradient Behavior of the Density Functional Exchange Energy. The Journal of Chemical Physics 1986, 85 (12), 7184–7187. https://doi.org/10.1063/1.451353. ↩
Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. Physical Review A 1988, 38 (6), 3098–3100. https://doi.org/10.1103/PhysRevA.38.3098. ↩
Perdew, J. P. Density-Functional Approximation for the Correlation Energy of the Inhomogeneous Electron Gas. Physical Review B 1986, 33 (12), 8822–8824. https://doi.org/10.1103/PhysRevB.33.8822. ↩↩
Perdew, J. P.; Burke, K.; Ernzerhof, M. Errata: Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. https://doi.org/10.1103/PhysRevLett.77.3865. ↩↩
Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1997, 78, 1396–1396. https://doi.org/10.1103/PhysRevLett.78.1396. ↩↩
Gill, P. M. W. A New Gradient-Corrected Exchange Functional. Molecular Physics 1996, 89 (2), 433–445. https://doi.org/10.1080/002689796173813. ↩
Handy, N. C.; Cohen, A. J. Left-Right Correlation Energy. Molecular Physics 2001, 99 (5), 403–412. https://doi.org/10.1080/00268970010018431. ↩
Adamo, C.; Barone, V. Exchange Functionals with Improved Long-Range Behavior and Adiabatic Connection Methods Without Adjustable Parameters: The mPW and mPW1PW Models. The Journal of Chemical Physics 1998, 108 (2), 664–675. https://doi.org/10.1063/1.475428. ↩
Zhao, Y.; Truhlar, D. G. Design of Density Functionals That Are Broadly Accurate for Thermochemistry, Thermochemical Kinetics, and Nonbonded Interactions. The Journal of Physical Chemistry A 2005, 109 (25), 5656–5667. https://doi.org/10.1021/jp050536c. ↩
Filatov, M.; Thiel, W. A New Gradient-Corrected Exchange-Correlation Density Functional. Molecular Physics 1997, 91 (5), 847–860. https://doi.org/10.1080/002689797170950. ↩↩↩
Filatov, M.; Thiel, W. A Nonlocal Correlation Energy Density Functional from a Coulomb Hole Model. International Journal of Quantum Chemistry 1997, 62 (6), 603–616. https://doi.org/10.1002/(sici)1097-461x(1997)62:6<603::aid-qua4>3.0.co;2-#. ↩↩↩
Hammer, B.; Hansen, L. B.; Nørskov, J. K. Improved Adsorption Energetics Within Density-Functional Theory Using Revised Perdew-Burke-Ernzerhof Functionals. Physical Review B 1999, 59 (11), 7413–7421. https://doi.org/10.1103/physrevb.59.7413. ↩↩
Zhang, Y.; Yang, W. Comment on Generalized Gradient Approximation Made Simple. Physical Review Letters 1998, 80 (4), 890–890. https://doi.org/10.1103/physrevlett.80.890. ↩↩
Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Development and Assessment of a New Hybrid Density Functional Model for Thermochemical Kinetics. The Journal of Physical Chemistry A 2004, 108 (14), 2715–2719. https://doi.org/10.1021/jp049908s. ↩↩↩↩↩↩
Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Physical Review B 1988, 37 (2), 785–789. https://doi.org/10.1103/PhysRevB.37.785. ↩
Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Atoms, Molecules, Solids, and Surfaces: Applications of the Generalized Gradient Approximation for Exchange and Correlation. Physical Review B 1992, 46 (11), 6671–6687. https://doi.org/10.1103/PhysRevB.46.6671. ↩
Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Erratum: Atoms, Molecules, Solids, and Surfaces: Applications of the Generalized Gradient Approximation for Exchange and Correlation. Physical Review B 1993, 48 (7), 4978–4978. https://doi.org/10.1103/PhysRevB.48.4978.2. ↩
Tsuneda, T.; Suzumura, T.; Hirao, K. A New One-Parameter Progressive Collesalvetti-Type Correlation Functional. The Journal of Chemical Physics 1999, 110 (22), 10664–10678. https://doi.org/10.1063/1.479012. ↩↩
Hamprecht, F. A.; Cohen, A. J.; Tozer, D. J.; Handy, N. C. Development and Assessment of New Exchange-Correlation Functionals. The Journal of Chemical Physics 1998, 109 (15), 6264–6271. https://doi.org/10.1063/1.477267. ↩↩
Boese, A. D.; Doltsinis, N. L.; Handy, N. C.; Sprik, M. New Generalized Gradient Approximation Functionals. The Journal of Chemical Physics 2000, 112 (4), 1670–1678. https://doi.org/10.1063/1.480732. ↩↩
Boese, A. D.; Martin, J. M. L.; Handy, N. C. The Role of the Basis Set: Assessing Density Functional Theory. The Journal of Chemical Physics 2003, 119 (6), 3005–3014. https://doi.org/10.1063/1.1589004. ↩
Boese, A. D.; Handy, N. C. A New Parametrization of Exchangecorrelation Generalized Gradient Approximation Functionals. The Journal of Chemical Physics 2001, 114 (13), 5497–5503. https://doi.org/10.1063/1.1347371. ↩
Cohen, A. J.; Handy, N. C. Assessment of Exchange Correlation Functionals. Chemical Physics Letters 2000, 316 (1-2), 160–166. https://doi.org/10.1016/s0009-2614(99)01273-7. ↩
Menconi, G.; Wilson, P. J.; Tozer, D. J. Emphasizing the Exchange-Correlation Potential in Functional Development. The Journal of Chemical Physics 2001, 114 (9), 3958–3967. https://doi.org/10.1063/1.1342776. ↩
Boese, A. D.; Chandra, A.; Martin, J. M. L.; Marx, D. From Ab Initio Quantum Chemistry to Molecular Dynamics: The Delicate Case of Hydrogen Bonding in Ammonia. The Journal of Chemical Physics 2003, 119 (12), 5965–5980. https://doi.org/10.1063/1.1599338. ↩
Tsuneda, T.; Suzumura, T.; Hirao, K. A Reexamination of Exchange Energy Functionals. The Journal of Chemical Physics 1999, 111 (13), 5656–5667. https://doi.org/10.1063/1.479954. ↩
Perdew, J. P.; Kurth, S.; Zupan, A.; Blaha, P. Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation. Physical Review Letters 1999, 82 (12), 2544–2547. https://doi.org/10.1103/physrevlett.82.2544. ↩↩
Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the Density Functional Ladder: Nonempirical Meta-Generalized Gradient Approximation Designed for Molecules and Solids. Physical Review Letters 2003, 91 (14), 146401. https://doi.org/10.1103/physrevlett.91.146401. ↩↩
Zhao, Y.; Truhlar, D. G. Hybrid Meta Density Functional Theory Methods for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions: The MPW1B95 and MPWB1K Models and Comparative Assessments for Hydrogen Bonding and van Der Waals Interactions. The Journal of Physical Chemistry A 2004, 108 (33), 6908–6918. https://doi.org/10.1021/jp048147q. ↩↩↩
Zhao, Y.; Schultz, N. E.; Truhlar, D. G. Exchange-Correlation Functional with Broad Accuracy for Metallic and Nonmetallic Compounds, Kinetics, and Noncovalent Interactions. The Journal of Chemical Physics 2005, 123 (16), 161103. https://doi.org/10.1063/1.2126975. ↩↩
Zhao, Y.; Schultz, N. E.; Truhlar, D. G. Design of Density Functionals by Combining the Method of Constraint Satisfaction with Parametrization for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions. Journal of Chemical Theory and Computation 2006, 2 (2), 364–382. https://doi.org/10.1021/ct0502763. ↩↩↩
Staroverov, V. N.; Scuseria, G. E.; Tao, J.; Perdew, J. P. Comparative Assessment of a New Nonempirical Density Functional: Molecules and Hydrogen-Bonded Complexes. The Journal of Chemical Physics 2003, 119 (23), 12129–12137. https://doi.org/10.1063/1.1626543. ↩
Becke, A. D. Density-Functional Thermochemistry. IV. A New Dynamical Correlation Functional and Implications for Exact-Exchange Mixing. The Journal of Chemical Physics 1996, 104 (3), 1040–1046. https://doi.org/10.1063/1.470829. ↩↩
Voorhis, T. V.; Scuseria, G. E. A Novel Form for the Exchange-Correlation Energy Functional. The Journal of Chemical Physics 1998, 109 (2), 400–410. https://doi.org/10.1063/1.476577. ↩↩↩
Zhao, Y.; Truhlar, D. G. A New Local Density Functional for Main-Group Thermochemistry, Transition Metal Bonding, Thermochemical Kinetics, and Noncovalent Interactions. The Journal of Chemical Physics 2006, 125 (19), 194101. https://doi.org/10.1063/1.2370993. ↩↩↩↩↩↩
Zhao, Y.; Truhlar, D. G. Density Functional for Spectroscopy: No Long-Range Self-Interaction Error, Good Performance for Rydberg and Charge-Transfer States, and Better Performance on Average Than B3LYP for Ground States. The Journal of Physical Chemistry A 2006, 110 (49), 13126–13130. https://doi.org/10.1021/jp066479k. ↩↩↩
Zhao, Y.; Truhlar, D. G. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals. Theoretical Chemistry Accounts 2008, 120 (1-3), 215–241. https://doi.org/10.1007/s00214-007-0310-x. ↩↩↩
Zhao, Y.; Truhlar, D. G. Construction of a Generalized Gradient Approximation by Restoring the Density-Gradient Expansion and Enforcing a Tight Lieboxford Bound. The Journal of Chemical Physics 2008, 128 (18), 184109. https://doi.org/10.1063/1.2912068. ↩↩↩↩↩↩↩↩↩
Peverati, R.; Truhlar, D. G. Improving the Accuracy of Hybrid Meta-GGA Density Functionals by Range Separation. The Journal of Physical Chemistry Letters 2011, 2 (21), 2810–2817. https://doi.org/10.1021/jz201170d. ↩↩↩
Peverati, R.; Truhlar, D. G. M11-l: A Local Density Functional That Provides Improved Accuracy for Electronic Structure Calculations in Chemistry and Physics. The Journal of Physical Chemistry Letters 2012, 3 (1), 117–124. https://doi.org/10.1021/jz201525m. ↩↩↩
Peverati, R.; Zhao, Y.; Truhlar, D. G. Generalized Gradient Approximation That Recovers the Second-Order Density-Gradient Expansion with Optimized Across-the-Board Performance. The Journal of Physical Chemistry Letters 2011, 2 (16), 1991–1997. https://doi.org/10.1021/jz200616w. ↩↩↩
Peverati, R.; Truhlar, D. G. Communication: A Global Hybrid Generalized Gradient Approximation to the Exchange-Correlation Functional That Satisfies the Second-Order Density-Gradient Constraint and Has Broad Applicability in Chemistry. The Journal of Chemical Physics 2011, 135 (19), 191102. https://doi.org/10.1063/1.3663871. ↩↩↩
Pernal, K.; Podeszwa, R.; Patkowski, K.; Szalewicz, K. Dispersionless Density Functional Theory. Physical Review Letters 2009, 103 (26), 263201. https://doi.org/10.1103/physrevlett.103.263201. ↩
Wilson, P. J.; Bradley, T. J.; Tozer, D. J. Hybrid Exchange-Correlation Functional Determined from Thermochemical Data and Ab Initio Potentials. The Journal of Chemical Physics 2001, 115 (20), 9233–9242. https://doi.org/10.1063/1.1412605. ↩
Keal, T. W.; Tozer, D. J. Semiempirical Hybrid Functional with Improved Performance in an Extensive Chemical Assessment. The Journal of Chemical Physics 2005, 123 (12), 121103. https://doi.org/10.1063/1.2061227. ↩
Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction. Journal of Computational Chemistry 2006, 27 (15), 1787–1799. https://doi.org/10.1002/jcc.20495. ↩
Schmider, H. L.; Becke, A. D. Optimized Density Functionals from the Extended G2 Test Set. The Journal of Chemical Physics 1998, 108 (23), 9624–9631. https://doi.org/10.1063/1.476438. ↩
Adamo, C.; Barone, V. Toward Reliable Density Functional Methods Without Adjustable Parameters: The PBE0 Model. The Journal of Chemical Physics 1999, 110 (13), 6158–6170. https://doi.org/10.1063/1.478522. ↩
Lynch, B. J.; Fast, P. L.; Harris, M.; Truhlar, D. G. Adiabatic Connection for Kinetics. The Journal of Physical Chemistry A 2000, 104 (21), 4811–4815. https://doi.org/10.1021/jp000497z. ↩
Sun, J.; Perdew, J. P.; Ruzsinszky, A. Semilocal Density Functional Obeying a Strongly Tightened Bound for Exchange. Proceedings of the National Academy of Sciences 2015, 112 (3), 685–689. https://doi.org/10.1073/pnas.1423145112. ↩
Verma, P.; Truhlar, D. G. HLE16: A Local Kohn-Sham Gradient Approximation with Good Performance for Semiconductor Band Gaps and Molecular Excitation Energies. The Journal of Physical Chemistry Letters 2017, 8 (2), 380–387. https://doi.org/10.1021/acs.jpclett.6b02757. ↩
Yang, Z.; Peng, H.; Sun, J.; Perdew, J. P. More Realistic Band Gaps from Meta-Generalized Gradient Approximations: Only in a Generalized Kohn-Sham Scheme. Physical Review B 2016, 93 (20), 205205. https://doi.org/10.1103/physrevb.93.205205. ↩
Mejia-Rodriguez, D.; Trickey, S. B. Deorbitalization Strategies for Meta-Generalized-Gradient-Approximation Exchange-Correlation Functionals. Physical Review A 2017, 96 (5), 052512. https://doi.org/10.1103/physreva.96.052512. ↩
Wang, Y.; Jin, X.; Yu, H. S.; Truhlar, D. G.; He, X. Revised M06-l Functional for Improved Accuracy on Chemical Reaction Barrier Heights, Noncovalent Interactions, and Solid-State Physics. Proceedings of the National Academy of Sciences 2017, 114 (32), 8487–8492. https://doi.org/10.1073/pnas.1705670114. ↩
Wang, Y.; Verma, P.; Jin, X.; Truhlar, D. G.; He, X. Revised M06 Density Functional for Main-Group and Transition-Metal Chemistry. Proceedings of the National Academy of Sciences 2018, 115 (41), 10257–10262. https://doi.org/10.1073/pnas.1810421115. ↩
Chai, J.-D.; Head-Gordon, M. Systematic Optimization of Long-Range Corrected Hybrid Density Functionals. The Journal of Chemical Physics 2008, 128 (8), 084106. https://doi.org/10.1063/1.2834918. ↩
Lin, Y.-S.; Li, G.-D.; Mao, S.-P.; Chai, J.-D. Long-Range Corrected Hybrid Density Functionals with Improved Dispersion Corrections. Journal of Chemical Theory and Computation 2012, 9 (1), 263–272. https://doi.org/10.1021/ct300715s. ↩
Bartók, A. P.; Yates, J. R. Regularized SCAN Functional. The Journal of Chemical Physics 2019, 150 (16), 161101. https://doi.org/10.1063/1.5094646. ↩
Furness, J. W.; Kaplan, A. D.; Ning, J.; Perdew, J. P.; Sun, J. Accurate and Numerically Efficient r2SCAN Meta-Generalized Gradient Approximation. The Journal of Physical Chemistry Letters 2020, 11 (19), 8208–8215. https://doi.org/10.1021/acs.jpclett.0c02405. ↩
Bursch, M.; Neugebauer, H.; Ehlert, S.; Grimme, S. Dispersion Corrected r2SCAN Based Global Hybrid Functionals: r2SCANh, r2SCAN0, and r2SCAN50. The Journal of Chemical Physics 2022, 156 (13), 134105. https://doi.org/10.1063/5.0086040. ↩
Mejía-Rodríguez, D.; Trickey, S. B. Meta-GGA Performance in Solids at Almost GGA Cost. Physical Review B 2020, 102 (12), 121109. https://doi.org/10.1103/physrevb.102.121109. ↩
Mejía-Rodríguez, D.; Trickey, S. B. Spin-Crossover from a Well-Behaved, Low-Cost Meta-GGA Density Functional. The Journal of Physical Chemistry A 2020, 124 (47), 9889–9894. https://doi.org/10.1021/acs.jpca.0c08883. ↩
Carmona-Espíndola, J.; Gázquez, J. L.; Vela, A.; Trickey, S. B. Generalized Gradient Approximation Exchange Energy Functional with Near-Best Semilocal Performance. Journal of Chemical Theory and Computation 2018, 15 (1), 303–310. https://doi.org/10.1021/acs.jctc.8b00998. ↩
Kurth, S.; Perdew, J. P.; Blaha, P. Molecular and Solid-State Tests of Density Functional Approximations: LSD, GGAs, and Meta-GGAs. International Journal of Quantum Chemistry 1999, 75 (4-5), 889–909. https://doi.org/10.1002/(sici)1097-461x(1999)75:4/5<889::aid-qua54>3.0.co;2-8. ↩
Savin, A. Beyond the Kohn-Sham Determinant. In Recent advances in density functional methods; WORLD SCIENTIFIC, 1995; pp 129–153. https://doi.org/10.1142/9789812830586\_0004. ↩
Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K. A Long-Range Correction Scheme for Generalized-Gradient-Approximation Exchange Functionals. The Journal of Chemical Physics 2001, 115 (8), 3540–3544. https://doi.org/10.1063/1.1383587. ↩
Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. A Long-Range-Corrected Time-Dependent Density Functional Theory. The Journal of Chemical Physics 2004, 120 (18), 8425–8433. https://doi.org/10.1063/1.1688752. ↩
Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid Exchange-Correlation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chemical Physics Letters 2004, 393 (1-3), 51–57. https://doi.org/10.1016/j.cplett.2004.06.011. ↩
Peach, M. J. G.; Cohen, A. J.; Tozer, D. J. Influence of Coulomb-Attenuation on Exchange-Correlation Functional Quality. Phys. Chem. Chem. Phys. 2006, 8 (39), 4543–4549. https://doi.org/10.1039/b608553a. ↩
Song, J.-W.; Hirosawa, T.; Tsuneda, T.; Hirao, K. Long-Range Corrected Density Functional Calculations of Chemical Reactions: Redetermination of Parameter. The Journal of Chemical Physics 2007, 126 (15), 154105. https://doi.org/10.1063/1.2721532. ↩
Livshits, E.; Baer, R. A Well-Tempered Density Functional Theory of Electrons in Molecules. Physical Chemistry Chemical Physics 2007, 9 (23), 2932. https://doi.org/10.1039/b617919c. ↩
Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Development of Exchange-Correlation Functionals with Minimal Many-Electron Self-Interaction Error. The Journal of Chemical Physics 2007, 126 (19), 191109. https://doi.org/10.1063/1.2741248. ↩
Rohrdanz, M. A.; Herbert, J. M. Simultaneous Benchmarking of Ground- and Excited-State Properties with Long-Range-Corrected Density Functional Theory. The Journal of Chemical Physics 2008, 129 (3), 034107. https://doi.org/10.1063/1.2954017. ↩
Govind, N.; Valiev, M.; Jensen, L.; Kowalski, K. Excitation Energies of Zinc Porphyrin in Aqueous Solution Using Long-Range Corrected Time-Dependent Density Functional Theory. The Journal of Physical Chemistry A 2009, 113 (21), 6041–6043. https://doi.org/10.1021/jp902118k. ↩
Baer, R.; Livshits, E.; Salzner, U. Tuned Range-Separated Hybrids in Density Functional Theory. Annual Review of Physical Chemistry 2010, 61 (1), 85–109. https://doi.org/10.1146/annurev.physchem.012809.103321. ↩
Autschbach, J.; Srebro, M. Delocalization Error and Functional Tuning in Kohn-Sham Calculations of Molecular Properties. Accounts of Chemical Research 2014, 47 (8), 2592–2602. https://doi.org/10.1021/ar500171t. ↩
Verma, P.; Bartlett, R. J. Increasing the Applicability of Density Functional Theory. IV. Consequences of Ionization-Potential Improved Exchange-Correlation Potentials. The Journal of Chemical Physics 2014, 140 (18), 18A534. https://doi.org/10.1063/1.4871409. ↩
Swart, M.; Solà, M.; Bickelhaupt, F. M. A New All-Round Density Functional Based on Spin States and SN2 Barriers. The Journal of Chemical Physics 2009, 131 (9), 094103. https://doi.org/10.1063/1.3213193. ↩
Swart, M.; Solà, M.; Bickelhaupt, F. M. Switching Between OPTX and PBE Exchange Functionals. Journal of Computational Methods in Sciences and Engineering 2009, 9 (1-2), 69–77. https://doi.org/10.3233/jcm-2009-0230. ↩
Grimme, S. Semiempirical Hybrid Density Functional with Perturbative Second-Order Correlation. The Journal of Chemical Physics 2006, 124 (3), 034108. https://doi.org/10.1063/1.2148954. ↩↩
Rabuck, A. D.; Scuseria, G. E. Improving Self-Consistent Field Convergence by Varying Occupation Numbers. The Journal of Chemical Physics 1999, 110 (2), 695–700. https://doi.org/10.1063/1.478177. ↩
Wu, Q.; Voorhis, T. V. Direct Optimization Method to Study Constrained Systems Within Density-Functional Theory. Physical Review A 2005, 72 (2), 024502. https://doi.org/10.1103/physreva.72.024502. ↩
Warren, R. W.; Dunlap, B. I. Fractional Occupation Numbers and Density Functional Energy Gradients Within the Linear Combination of Gaussian-Type Orbitals Approach. Chemical Physics Letters 1996, 262 (3-4), 384–392. https://doi.org/10.1016/0009-2614(96)01107-4. ↩
Becke, A. D. A Multicenter Numerical Integration Scheme for Polyatomic Molecules. The Journal of Chemical Physics 1988, 88 (4), 2547–2553. https://doi.org/10.1063/1.454033. ↩
Stratmann, R. E.; Scuseria, G. E.; Frisch, M. J. Achieving Linear Scaling in Exchange-Correlation Density Functional Quadratures. Chemical Physics Letters 1996, 257 (3-4), 213–223. https://doi.org/10.1016/0009-2614(96)00600-8. ↩
Murray, C. W.; Handy, N. C.; Laming, G. J. Quadrature Schemes for Integrals of Density Functional Theory. Molecular Physics 1993, 78 (4), 997–1014. https://doi.org/10.1080/00268979300100651. ↩
Mura, M. E.; Knowles, P. J. Improved Radial Grids for Quadrature in Molecular Density-Functional Calculations. The Journal of Chemical Physics 1996, 104 (24), 9848–9858. https://doi.org/10.1063/1.471749. ↩
Treutler, O.; Ahlrichs, R. Efficient Molecular Numerical Integration Schemes. The Journal of Chemical Physics 1995, 102 (1), 346–354. https://doi.org/10.1063/1.469408. ↩
Sharp, R. T.; Horton, G. K. A Variational Approach to the Unipotential Many-Electron Problem. Physical Review 1953, 90 (2), 317–317. https://doi.org/10.1103/physrev.90.317. ↩
Talman, J. D.; Shadwick, W. F. Optimized Effective Atomic Central Potential. Physical Review A 1976, 14 (1), 36–40. https://doi.org/10.1103/physreva.14.36. ↩
Krieger, J. B.; Li, Y.; Iafrate, G. J. Construction and Application of an Accurate Local Spin-Polarized Kohn-Sham Potential with Integer Discontinuity: Exchange-Only Theory. Physical Review A 1992, 45 (1), 101–126. https://doi.org/10.1103/physreva.45.101. ↩
Krieger, J. B.; Li, Y.; Iafrate, G. J. Systematic Approximations to the Optimized Effective Potential: Application to Orbital-Density-Functional Theory. Physical Review A 1992, 46 (9), 5453–5458. https://doi.org/10.1103/physreva.46.5453. ↩
Li, Y.; Krieger, J. B.; Iafrate, G. J. Self-Consistent Calculations of Atomic Properties Using Self-Interaction-Free Exchange-Only Kohn-Sham Potentials. Physical Review A 1993, 47 (1), 165–181. https://doi.org/10.1103/physreva.47.165. ↩
Garza, J.; Nichols, J. A.; Dixon, D. A. The Optimized Effective Potential and the Self-Interaction Correction in Density Functional Theory: Application to Molecules. The Journal of Chemical Physics 2000, 112 (18), 7880–7890. https://doi.org/10.1063/1.481421. ↩
Galvan, M.; Vela, A.; Gazquez, J. L. Chemical Reactivity in Spin-Polarized Density Functional Theory. The Journal of Physical Chemistry 1988, 92 (22), 6470–6474. https://doi.org/10.1021/j100333a056. ↩
Chamorro, E.; Pérez, P. Condensed-to-Atoms Electronic Fukui Functions Within the Framework of Spin-Polarized Density-Functional Theory. The Journal of Chemical Physics 2005, 123 (11), 114107. https://doi.org/10.1063/1.2033689. ↩
Wu, Q.; Yang, W. Empirical Correction to Density Functional Theory for van Der Waals Interactions. The Journal of Chemical Physics 2002, 116 (2), 515–524. https://doi.org/10.1063/1.1424928. ↩
Zimmerli, U.; Parrinello, M.; Koumoutsakos, P. Dispersion Corrections to Density Functionals for Water Aromatic Interactions. The Journal of Chemical Physics 2004, 120 (6), 2693–2699. https://doi.org/10.1063/1.1637034. ↩
Grimme, S. Accurate Description of van Der Waals Complexes by Density Functional Theory Including Empirical Corrections. Journal of Computational Chemistry 2004, 25 (12), 1463–1473. https://doi.org/10.1002/jcc.20078. ↩
Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-d) for the 94 Elements h-Pu. The Journal of Chemical Physics 2010, 132 (15), 154104. https://doi.org/10.1063/1.3382344. ↩
Otero-de-la-Roza, A.; Johnson, E. R. Non-Covalent Interactions and Thermochemistry Using XDM-Corrected Hybrid and Range-Separated Hybrid Density Functionals. The Journal of Chemical Physics 2013, 138 (20), 204109. https://doi.org/10.1063/1.4807330. ↩