5.6 Range-Separated Hybrid Density Functionals

5.6.2 User-Defined RSH Functionals

As pointed out in Ref. Dreuw:2003 and elsewhere, the description of charge-transfer excited states within density functional theory (or more precisely, time-dependent DFT, which is discussed in Section 7.3) requires full (100%) non-local HF exchange, at least in the limit of large donor–acceptor distance. Hybrid functionals such as B3LYPBecke:1993b, Stephens:1994 and PBE0Adamo:1999b that are well-established and in widespread use, however, employ only 20% and 25% HF exchange, respectively. While these functionals provide excellent results for many ground-state properties, they cannot correctly describe the distance dependence of charge-transfer excitation energies, which are enormously underestimated by most common density functionals. This is a serious problem in any case, but it is a catastrophic problem in large molecules and in non-covalent clusters, where TDDFT often predicts a near-continuum of spurious, low-lying charge transfer states.Lange:2007, Lange:2009 The problems with TDDFT’s description of charge transfer are not limited to large donor–acceptor distances, but have been observed at 2 Å separation, in systems as small as uracil–(H2O)4.Lange:2007 Rydberg excitation energies also tend to be substantially underestimated by standard TDDFT.

One possible avenue by which to correct such problems is to parameterize functionals that contain 100% HF exchange, though few such functionals exist to date. An alternative option is to attempt to preserve the form of common GGAs and hybrid functionals at short range (i.e., keep the 25% HF exchange in PBE0) while incorporating 100% HF exchange at long range, which provides a rigorously correct description of the long-range distance dependence of charge-transfer excitation energies, but aims to avoid contaminating short-range exchange-correlation effects with additional HF exchange. The separation is accomplished using the range-separation ansatz that was introduced in Section 5.3. In particular, functionals that use 100% HF exchange at long range (cx,LR=1 in Eq. (5.13)) are known as “long-range-corrected” (LRC) functionals. An LRC version of PBE0 would, for example, have cx,SR=0.25.

To fully specify an LRC functional, one must choose a value for the range separation parameter ω in Eq. (5.12). In the limit ω0, the LRC functional in Eq. (5.13) reduces to a non-RSH functional where there is no “SR” or “LR”, because all exchange and correlation energies are evaluated using the full Coulomb operator, r12-1. Meanwhile the ω limit corresponds to a new functional, ExcRSH=Ec+ExHF. Full HF exchange is inappropriate for use with most contemporary GGA correlation functionals, so the latter limit is expected to perform quite poorly. Values of ω>1.0 bohr-1 are likely not worth considering, according to benchmark tests.Lange:2008, Rohrdanz:2008

Evaluation of the short- and long-range HF exchange energies is straightforward,Adamson:1999 so the crux of any RSH functional is the form of the short-range GGA exchange functional, and several such functionals are available in Q-Chem. These include short-range variants of the B88 and PBE exchange described by Hirao and co-workers,Iikura:2001, Song:2007 called μB88 and μPBE in Q-Chem,Richard:2011 and an alternative formulation of short-range PBE exchange proposed by Scuseria and co-workers,Henderson:2008 which is known as ωPBE. These functionals are available in Q-Chem thanks to the efforts of the Herbert group.Rohrdanz:2008, Rohrdanz:2009 By way of notation, the terms “μPBE”, “ωPBE”, etc., refer only to the short-range exchange functional, Ex,SRDFT in Eq. (5.13). These functionals could be used in “screened exchange” mode, as described in Section 5.3, as for example in the HSE03 functional,Heyd:2003 therefore the designation “LRC-ωPBE”, for example, should only be used when the short-range exchange functional ωPBE is combined with 100% Hartree-Fock exchange in the long range.

In general, LRC-DFT functionals have been shown to remove the near-continuum of spurious charge-transfer excited states that appear in large-scale TDDFT calculations.Lange:2008 However, certain results depend sensitively upon the value of the range-separation parameter ω,Lange:2008, Rohrdanz:2008, Rohrdanz:2009, Lange:2009, Uhlig:2014 especially in TDDFT calculations (Section 7.3) and therefore the results of LRC-DFT calculations must therefore be interpreted with caution, and probably for a range of ω values. This can be accomplished by requesting a functional that contains some short-range GGA exchange functional (ωPBE or μPBE, in the examples mentioned above), in combination with setting the $rem variable LRC_DFT = TRUE, which requests the addition of 100% Hartree-Fock exchange in the long-range. Basic job-control variables and an example can be found below. The value of the range-separation parameter is then controlled by the variable OMEGA, as shown in the examples below.

LRC_DFT
       Controls the application of long-range-corrected DFT
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       FALSE (or 0) Do not apply long-range correction. TRUE (or 1) Add 100% long-range Hartree-Fock exchange to the requested functional.
RECOMMENDATION:
       The $rem variable OMEGA must also be specified, in order to set the range-separation parameter.

OMEGA
       Sets the range-separation parameter, ω, also known as μ, in functionals based on Hirao’s RSH scheme.
TYPE:
       INTEGER
DEFAULT:
       No default
OPTIONS:
       n Corresponding to ω=n/1000, in units of bohr-1
RECOMMENDATION:
       None

COMBINE_K
       Controls separate or combined builds for short-range and long-range K
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       FALSE (or 0) Build short-range and long-range K separately (twice as expensive as a global hybrid) TRUE (or 1) Build short-range and long-range K together ( as expensive as a global hybrid)
RECOMMENDATION:
       Most pre-defined range-separated hybrid functionals in Q-Chem use this feature by default. However, if a user-specified RSH is desired, it is necessary to manually turn this feature on.

HFK_SR_COEF
       Sets the coefficient for short-range HF exchange
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       n Corresponding to n/100000000
RECOMMENDATION:
       None

HFK_LR_COEF
       Sets the coefficient for long-range HF exchange
TYPE:
       INTEGER
DEFAULT:
       100000000
OPTIONS:
       n Corresponding to n/100000000
RECOMMENDATION:
       None

Example 5.6  Application of LRC-BOP to (H2O)2-.

$comment
The value of omega is 0.47 by default but can
be overwritten by specifying OMEGA.
$end

$molecule
-1 2
O           1.347338    -0.017773    -0.071860
H           1.824285     0.813088     0.117645
H           1.805176    -0.695567     0.461913
O          -1.523051    -0.002159    -0.090765
H          -0.544777    -0.024370    -0.165445
H          -1.682218     0.174228     0.849364
$end

$rem
   EXCHANGE      LRC-BOP
   BASIS         6-31(1+,3+)G*
   LRC_DFT       TRUE
   OMEGA         300      ! = 0.300 bohr**(-1)
$end

Rohrdanz et al.Rohrdanz:2009 published a thorough benchmark study of both ground- and excited-state properties using the LRC-ωPBEh functional, in which the “h” indicates a short-range hybrid (i.e., the presence of some short-range HF exchange). Empirically-optimized parameters of cx,SR=0.2 (see Eq. (5.13)) and ω=0.2 bohr-1 were obtained,Rohrdanz:2009 and these parameters are taken as the defaults for LRC-ωPBEh. Caution is warranted, however, especially in TDDFT calculations for large systems, as excitation energies for states that exhibit charge-transfer character can be rather sensitive to the precise value of ω.Lange:2009, Rohrdanz:2009 In such cases (and maybe in general), the “tuning” procedure described in Section 5.6.3 is recommended.

Example 5.7  Application of LRC-ωPBEh to the C2H4C2F4 dimer at 5 Å separation.

$comment
    This example uses the "optimal" parameter set discussed above.
    It can also be run by setting METHOD = LRC-wPBEh.
$end

$molecule
0 1
C           0.670604    0.000000    0.000000
C          -0.670604    0.000000    0.000000
H           1.249222    0.929447    0.000000
H           1.249222   -0.929447    0.000000
H          -1.249222    0.929447    0.000000
H          -1.249222   -0.929447    0.000000
C           0.669726    0.000000    5.000000
C          -0.669726    0.000000    5.000000
F           1.401152    1.122634    5.000000
F           1.401152   -1.122634    5.000000
F          -1.401152   -1.122634    5.000000
F          -1.401152    1.122634    5.000000
$end

$rem
   EXCHANGE      GEN
   BASIS         6-31+G*
   LRC_DFT       TRUE
   OMEGA         200      ! = 0.2 a.u.
   CIS_N_ROOTS   4
   CIS_TRIPLETS  FALSE
$end

$xc_functional
   C  PBE   1.00
   X  wPBE  0.80
   X  HF    0.20
$end

Both LRC functionals and also the asymptotic corrections that will be discussed in Section 5.10.1 are thought to reduce self-interaction error in approximate DFT. A convenient way to quantify—or at least depict—this error is by plotting the DFT energy as a function of the (fractional) number of electrons, N, because E(N) should in principle consist of a sequence of line segments with abrupt changes in slope (the so-called derivative discontinuityCohen:2008, Mori-Sanchez:2014) at integer values of N, but in practice these E(N) plots bow away from straight-line segments.Cohen:2008 Examination of such plots has been suggested as a means to adjust the fraction of short-range exchange in an LRC functional,Autschbach:2014 while the range-separation parameter is tuned as described in Section 5.6.3.

FRACTIONAL_ELECTRON
       Add or subtract a fraction of an electron.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Use an integer number of electrons. n Add n/1000 electrons to the system.
RECOMMENDATION:
       Use only if trying to generate E(N) plots. If n<0, a fraction of an electron is removed from the system.

Example 5.8  Example of a DFT job with a fractional number of electrons. Here, we make the -1.x anion of fluoride by subtracting a fraction of an electron from the HOMO of F2-.

$comment
   Subtracting a whole electron recovers the energy of F-.
   Adding electrons to the LUMO is possible as well.
$end

$rem
   EXCHANGE              b3lyp
   BASIS                 6-31+G*
   FRACTIONAL_ELECTRON  -500   ! divide by 1000 to get the fraction, -0.5 here.
   GEN_SCFMAN            FALSE ! not yet available in new scf code
$end

$molecule
-2 2
F
$end