Rigorous formulation of two-parameter double-hybrid density-functionals

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by  Emmanuel Fromager

Rigorous formulation of two-parameter double-hybrid density-functionals
Emmanuel Fromager
Laboratoire de Chimie Quantique,
Institut de Chimie, CNRS / Universit´e de Strasbourg,
4 rue Blaise Pascal, 67000 Strasbourg, France.
A two-parameter extension of the density-scaled double hybrid approach of Sharkas et al. [J.
Chem. Phys. 134, 064113 (2011)] is presented. It is based on the explicit treatment of a fraction
of multideterminantal exact exchange. The connection with conventional double hybrids is made
2 when neglecting density scaling in the correlation functional as well as second-order corrections to
1 thedensity. Inthiscontext,thefractionac ofsecond-orderMøller-Plesset(MP2)correlationenergy
0 isnotnecessarilyequaltothesquareofthefractionax ofHartree-Fockexchange. Morespecifically,
2 it is shown that ac ≤ a2x, a condition that conventional semi-empirical double hybrids actually
fulfill. In addition, a new procedure for calculating the orbitals, which has a better justification
n
a than the one routinely used, is proposed. Referred to as λ1 variant, the corresponding double
J hybrid approximation has been tested on a small set consisting of H2, N2, Be2, Mg2 and Ar2.
Three conventional double hybrids (B2-PLYP, B2GP-PLYP and PBE0-DH) have been considered.
1
1 Potentialcurvesobtainedwithλ1-andregulardoublehybridscan,insomecases,differsignificantly.
Inparticular,fortheweaklybounddimers,theλ1variantsbindsystematicallymorethantheregular
ones,whichisanimprovementinmanybutnotallcases. Includingdensityscalinginthecorrelation
]
h functionals may of course change the results significantly. Moreover, optimized effective potentials
p (OEPs)basedonapartially-interactingsystemcouldalsobeusedtogenerateproperorbitals. Work
- is currently in progress in those directions.
m
e
h I. INTRODUCTION and λ -double hybrids on a small test set consisting of
1
c H , N , Be , Mg and Ar are presented and discussed
2 2 2 2 2
.
s in Sec. IV.
c Sharkas et al.? recently proposed a rigorous formula-
i tion of one-parameter double hybrid density-functionals,
s
y which is based on the combination of second-order
II. THEORY
h Møller-Plesset (MP2) perturbation theory with a
p density-scaled correlation functional. The connection
[ with regular double hybrids? could be made when A. Multideterminantal exact exchange

2 neglecting density scaling. Note that, in their approach,
v the fraction ac of MP2 correlation energy equals the TheapproachrecentlyproposedbySharkaset al.? for
5 squareofthefractionax ofHartree-Fock(HF)exchange. deriving rigorous one-parameter double-hybrid function-
8
Such a condition is actually not fulfilled by conventional alsisbasedontheseparationoftheuniversalHohenberg-
4
1 double hybrids. The ac parameter is either chosen Kohn? functional F[n] = Fλ1[n] + EλH1xc[n] into a
irrespective of a or expressed in terms of a but not
. x x partially-interacting contribution
9 as a = a2. In the latter case, one can for example
c x
0
1 r(PefBerEt0o-DthHe)Pfuenrdcetiwo-nBaulrokfeB-Er´ermnzoenrhdoaf-nzderAoddaomubo?le,hwyhbircihd Fλ1[n]=Ψm→inn(cid:104)Ψ|Tˆ+λ1Wˆee|Ψ(cid:105)
1 (1)
v: is characterized by ac =a3x. =(cid:104)Ψλ1[n]|Tˆ+λ1Wˆee|Ψλ1[n](cid:105),
i
X
This work deals with the rigorous formulation of
with 0 ≤ λ ≤ 1, and the complement λ -dependent
1 1
r density-scaled two-parameter double hybrids. It is or-
a Hartree-exchange-correlation (Hxc) density-functional
ganized as follows: an exact two-parameter energy ex-
pression is firstly derived in Sec. IIA. Approximate for-
mulations are then investigated for defining single hy- Eλ1 [n]=(1−λ )E [n]+Eλ1[n],
brid (Sec. IIB) and double hybrid (Sec. IIC) energy ex- Hxc 1 Hx c
(2)
pressions. In Sec. IID, the connection with conventional
double hybrids is made. In addition, a new procedure Eλc1[n]=Ec[n]−Ecλ1[n],
for computing orbitals is proposed, defining thus what
is referred to as λ variant of the double hybrids. A where E [n] = (cid:104)ΦKS[n]|Wˆ |ΦKS[n](cid:105) is the usual exact
1 Hx ee
summary of the different approximations that have been Hartree-exchange(Hx)termbasedonthenon-interacting
formulated is then given in Sec. IIE. Following the com- Kohn-Sham (KS) determinant? . The correlation func-
putationaldetails(Sec.III),resultsobtainedwithregular tionalsEc[n]andEcλ1[n]correspondtofullyandpartially
2
λ -interacting systems, respectively. The exact ground- andintroducethecomplementλ -andλ -dependentHxc
1 1 2
state energy is then expressed as follows, according to density-functional energy:
the variational principle? ,
(cid:110) EλH1x,cλ2[n]=(1−λ2)(cid:104)Ψλ1[n]|Wˆee|Ψλ1[n](cid:105)+Eλc,1md[n]
E =min (cid:104)Ψλ1[n]|Tˆ+λ Wˆ +Vˆ |Ψλ1[n](cid:105)
1 ee ne
n +Eλ1 [n](cid:111), (3) =−(λ2−λ1)(cid:104)Ψλ1[n]|Wˆee|Ψλ1[n](cid:105) (10)
Hxc
where Vˆ =(cid:82) drv (r)nˆ(r) is the nuclear potential op- +EλH1x,md[n]+Eλc,1md[n],
ne ne
erator. Rewritten in terms of a minimization over wave
which, accordingtoEqs.(6)and(8), canberewrittenas
functions, Eq. (3) becomes
E =min(cid:110)(cid:104)Ψ|Tˆ+λ Wˆ +Vˆ |Ψ(cid:105)+Eλ1 [n ](cid:111) EλH1x,cλ2[n]=−(λ2−λ1)(cid:104)Ψλ1[n]|Wˆee|Ψλ1[n](cid:105)+EλH1xc[n]
1 ee ne Hxc Ψ
Ψ
(4) =(1−λ )E [n]+Eλ1[n] (11)
2 Hx c
=(cid:104)Ψλ1|Tˆ+λ1Wˆee+Vˆne|Ψλ1(cid:105)+EλH1xc[nΨλ1],
+(λ1−λ2)∆λc1[n].
where Ψλ1 fulfills the self-consistent equation
Usingthefollowingexpressionsbasedonauniformcoor-
(cid:18)Tˆ+λ Wˆ +Vˆ +Vˆλ1 [n ](cid:19)|Ψλ1(cid:105)=Eλ1|Ψλ1(cid:105), dinate scaling of the density? ? :
1 ee ne Hxc Ψλ1
(5) Ecλ1[n]=λ21Ec[n1/λ1],
VˆλH1xc[n]=(cid:90) dr δδEn(λHr1x)c[n]nˆ(r). ∆λc1[n]= ∂E∂cλλ[n](cid:12)(cid:12)(cid:12)(cid:12) , (12)
λ=λ1
Inordertointroduceasecondscalingfactorλ ,andthus
2
derive two-parameter double hybrids, let us consider the n1/λ1(r)=(1/λ1)3n(r/λ1),
following partitioning of the complement λ -Hxc func-
1
as well as Eq. (2), the more explicit density-scaled two-
tional:
parameter (DS2) form
Eλ1 [n]=Eλ1 [n]+Eλ1 [n], (6)
Hxc Hx,md c,md Eλ1,λ2[n]=(1−λ )E [n]+E [n]−λ2E [n ]
Hxc 2 Hx c 1 c 1/λ1
whichisbasedonthemultideterminantaldefinitionofthe
exact exchange (mdEXX), as introduced by Toulouse, +2λ (λ −λ )E [n ]
Gori-Giorgi and Savin ? ? ? in the context of range- 1 1 2 c 1/λ1 (13)
separated density-functional theory (DFT), +λ21(λ1−λ2)∂Ec∂[nλ1/λ](cid:12)(cid:12)(cid:12)(cid:12) ,
Eλ1 [n]=(1−λ )(cid:104)Ψλ1[n]|Wˆ |Ψλ1[n](cid:105). (7) λ=λ1
Hx,md 1 ee
is obtained and the exact ground-state energy in Eq. (3)
Thecorrespondingcomplementcorrelationfunctionaldif- can then be rewritten as follows:
fersthereforefromtheonegiveninEq.(2): accordingto
(cid:110)
Eqs. (6) and (7) it can be expressed as E =min (cid:104)Ψλ1[n]|Tˆ+λ Wˆ +Vˆ |Ψλ1[n](cid:105)
2 ee ne
n (cid:111) (14)
Eλ1 [n]=Eλ1[n]−(1−λ )∆λ1[n], +EλH1x,cλ2[n] ,
c,md c 1 c
(cid:16)
∆λ1[n]= (cid:104)Ψλ1[n]|Wˆ |Ψλ1[n](cid:105) (8) or, in terms of minimization over local potentials,
c ee
−(cid:104)ΦKS[n]|Wˆee|ΦKS[n](cid:105)(cid:17). E =min(cid:110)(cid:104)Ψλ1[v]|Tˆ+λ2Wˆee+Vˆne|Ψλ1[v](cid:105)
v (cid:111) (15)
+Eλ1,λ2[n ] ,
Recombined with Eq. (3), Eq. (7) leads to the explicit Hxc Ψλ1[v]
calculationof100%ofmdEXX.Inthespiritofusualhy-
brid functionals, we want to keep the flexibility of treat- where Ψλ1[v] denotes the ground state of the λ1-
ing only a fraction λ (0≤λ ≤1) of mdEXX explicitly. interactingHamiltonianTˆ+λ Wˆ +(cid:82) drv(r)nˆ(r). Itis
2 2 1 ee
For that purpose, we split the complement mdHx func- important to notice that, for a fixed λ value, the exact
1
tional as follows: minimizing potential vλ1 is the one which ensures that
thedensityofΨλ1[vλ1]equalstheexactground-stateden-
EλH1x,md[n]=(λ2−λ1)(cid:104)Ψλ1[n]|Wˆee|Ψλ1[n](cid:105) sity. Therefore,itdoesnotdependonλ2. Whenchoosing
(9) λ =λ ,theminimizationoverpotentialsinEq.(15)can
2 1
+(1−λ2)(cid:104)Ψλ1[n]|Wˆee|Ψλ1[n](cid:105), bereplacedbyaminimizationoverwavefunctionssothat
3
Eq. (4) is recovered and, according to Eq. (5), the exact Work is currently in progress in this direction. If we
potential vλ1 can be expressed as simply denote Φ and n the converged DS2-HF-OEP de-
terminantanditsdensity,theenergycanbereexpressed,
δEλ1 according to Eq. (13), as
vλ1(r)=v (r)+ Hxc[n ]. (16)
ne δn(r) Ψλ1
As a result, for any λ and λ values, the exact ground- EDS2,λ1,λ2 =(cid:104)Φ|Tˆ+Vˆ |Φ(cid:105)+E [n]+λ EHF[Φ]
1 2 HF-OEP ne H 2 x
state energy can be written as
E =(cid:104)Ψλ1|Tˆ+λ Wˆ +Vˆ |Ψλ1(cid:105)+Eλ1,λ2[n ]. (17) +(1−λ2)Ex[n]+Ec[n]−λ1(2λ2−λ1)Ec[n1/λ1] (20)
If λ2 (cid:54)= λ1, the 2eneeregy cannenot be obHtxacinedΨλs1traight- +λ21(λ1−λ2)∂Ec∂[nλ1/λ](cid:12)(cid:12)(cid:12)(cid:12) ,
fowardly from a minimization over wave functions: λ=λ1
(cid:110) whereEHF[Φ]istheHFexchangeenergy. Wethusobtain
E (cid:54)=min (cid:104)Ψ|Tˆ+λ2Wˆee+Vˆne|Ψ(cid:105) a densityx-scaled two-parameter hybrid (DS2H) exchange-
Ψ (cid:111) (18)
+Eλ1,λ2[n ] , correlation energy:
Hxc Ψ
since the minimizing wave function would be eigen- Eλ1,λ2 =λ EHF[Φ]+(1−λ )E [n]+E [n]
function of a λ -interacting system and therefore could xc,DS2H 2 x 2 x c
2
ninotterbacetienqguaslysttoemΨ.λ1A, swdhiiscchusissedeigiennfSuencc.tiIoInB,ofitaisλi1n- −λ1(2λ2−λ1)Ec[n1/λ1] (21)
pproitnecnitpilael p(OosEsiPb)lemtoethaoddaspttoregthuilsarcoonptteimxtiziend oerffdeecrtivtoe +λ21(λ1−λ2)∂Ec∂[nλ1/λ](cid:12)(cid:12)(cid:12)(cid:12) .
implement Eq. (15). λ=λ1
Notethat, intheparticularcaseλ =λ , theexactDS2-
1 2
HF-OEP potential can be expressed in terms of a func-
tional derivative like in the exact theory (see Sec. IIA).
B. Density-scaled two-parameter single hybrids
Indeed, the DS2-Hxc functional in Eq. (13) reduces then
to a DS1-Hxc functional which corresponds to the com-
This section deals with the formulation of single hy-
plement λ -Hxc functional:
1
brid functionals based on the energy expression given in
Eq. (15). The latter can be considered as a generaliza- Eλ1,λ1[n]=(1−λ )E [n]+E [n]−λ2E [n ]
tion of the standard OEP approach, which is based on Hxc 1 Hx c 1 c 1/λ1
(22)
a non-interacting KS system, since Ψλ1[v] is the ground =Eλ1 [n].
state of a partially-interacting system. In principle, any Hxc
wave-function-theory-basedmodelcanthusbecombined
Moreover, using the one-to-one correspondance between
rigorously with OEPs. We consider in this work the op-
thedensityn ofanysingledeterminantΦandthelocal
timization of the potential using the HF approximation Φ
Φλ1[v] to the exact wave function Ψλ1[v]. Following the potential vλ1[nΦ] such that the density of the HF deter-
terminologyofSharkaset al.? ,werefertothisapproach minantΦλ1[vλ1[nΦ]]equalsnΦ,theenergyinEq.(19)can
be simply obtained by minimization over single determi-
asDS2-HF-OEPapproximation. Inthiscase, theenergy
nants, that is without using OEPs. It then corresponds
becomes
to the DS1H energy of Sharkas et al.? which equals
EHDFS-2O,λE1P,λ2 =mvinEHDFS-2O,λE1P,λ2[v], EDλS11H =mΦin(cid:110)(cid:104)Φ|Tˆ+λ1Wˆee+Vˆne|Φ(cid:105)+EλH1xc[nΦ](cid:111).(23)
(cid:110)
EDS2,λ1,λ2[v]= (cid:104)Φλ1[v]|Tˆ+λ Wˆ +Vˆ |Φλ1[v](cid:105)
HF-OEP 2 ee ne (19) The minimizing determinant Φλ1 fulfills the self-
consistent equation
(cid:111)
+Eλ1,λ2[n ] .
Hxc Φλ1[v] (cid:18)Tˆ+λ Uˆ [Φλ1]+Vˆ +Vˆλ1 [n ](cid:19)|Φλ1(cid:105)=Eλ1 |Φλ1(cid:105),
1 HF ne Hxc Φλ1 DS1H
Such an approach can be interpreted as an exchange- (24)
only-typeOEPcalculationbasedonaλ1-interactingsys- where UˆHF[Φλ1] is the nonlocal HF potential, so that, in
tem. As a result, the DS2-HF-OEP energy gradient this particular case, the minimizing potential in Eq. (19)
has no simple analytical expression, like in the KS-OEP can be expressed as
scheme? . However,itcanbeefficientlycomputedfroma
HF-typelinearresponsevector,usingadecompositionof δEλ1
vλ1 (r)=v (r)+ Hxc[n ], (25)
the potential in an auxiliary basis of gaussian functions. DS1H ne δn(r) Φλ1
4
and the exchange-correlation in Eq. (21) reduces to we obtain the final energy expression through second or-
der:
Exλc1,DS1H =λ1ExHF[Φ]+(1−λ1)Ex[n]+Ec[n] (26) E0λ1,λ2 =(cid:104)Φλ01|Tˆ+λ2Wˆee+Vˆne|Φλ01(cid:105)
−λ21Ec[n1/λ1]. +EλH1x,cλ2[nΦλ1]+λ1(2λ2−λ1)EM(2P)
0
(cid:90) (cid:32) δEλ1,λ2 (cid:33) (33)
C. Density-scaled two-parameter double hybrids + dr v (r)−vλ1(r)+ Hxc [n ] δn(2)(r)
ne 0 δn(r) Φλ01
This section deals with the formulation of double hy- +...,
brid functionals based on the energy expression given in
Eq.(15). Inthefollowing,wedenotevλ1 any approxima-
0
tion to the exact minimizing potential vλ1. An approxi- whereδn(2)(r)denotesthesecond-ordercorrectiontothe
mate expression for the energy is thus obtained: density associated to Ψλ1. If we simply denote Φ and n
0
thezeroth-orderdeterminantΦλ1 anditsdensity,respec-
E0λ1,λ2 =(cid:104)Ψλ01|Tˆ+λ2Wˆee+Vˆne|Ψλ01(cid:105)+EλH1x,cλ2[nΨλ1],(27) tively, the exchange-correlation0energy has, according to
0
Eq.(13),theformofadensity scaled two-parameter dou-
where Ψλ1 is the ground state of the λ -interacting sys- ble hybrid (DS2DH) functional:
0 1
tem defined by vλ1:
0
(cid:16)Tˆ+λ1Wˆee+Vˆ0λ1(cid:17)|Ψλ01(cid:105)=E0λ1|Ψλ01(cid:105), (28) Exλc1,,DλS22DH =λ2ExHF[Φ]+(1−λ2)Ex[n]+Ec[n]
with Vˆλ1 =(cid:82) drvλ1(r)nˆ(r). The energy in Eq. (27) can −λ1(2λ2−λ1)Ec[n1/λ1]
0 0
be rewrEit0λt1e,nλ2a=s E0λ1 +(λ2−λ1)(cid:104)Ψλ01|Wˆee|Ψλ01(cid:105) +λ21(λ1−λ2)∂Ec∂[nλ1/λ](cid:12)(cid:12)(cid:12)(cid:12)λ=λ1 (34)
+(cid:104)Ψλ01|Vˆne−Vˆ0λ1|Ψλ01(cid:105) (29) +λ1(2λ2−λ1)EM(2P)
+Eλ1,λ2[n ], (cid:90) (cid:32) δEλ1,λ2 (cid:33)
Hxc Ψλ1 + dr v (r)−vλ1(r)+ Hxc [n] δn(2)(r).
0 ne 0 δn(r)
which is convenient when applying, as we propose in the
following, MP perturbation theory to the λ1-interacting Letusconsidertheparticularcaseλ1 =λ2. Accordingto
system described by Eq. (28). The zeroth-order wave Eqs.(22)and(25),whenchoosingtheDS1Hdeterminant
functionischosentobethedeterminantΦλ01 whichfulfills Φλ1 and potential vDλS11H as Φ and v0λ1, the second-order-
the HF-type equation density-correctiontermintheright-handsideofEq.(34)
cancels out and the DS1DH exchange-correlation energy
(cid:16) (cid:17)
Tˆ+λ1UˆHF[Φλ01]+Vˆ0λ1 |Φλ01(cid:105)=EHλ1F|Φλ01(cid:105). (30) of Sharkas et al.? is recovered:
Note that the Brillouin theorem is fulfilled in this con-
text, which means that the wave function contains only Exλc1,DS1DH =λ1ExHF[Φ]+(1−λ1)Ex[n]+Ec[n] (35)
double excitations through first order and, as a result,
thedensityremainsunchangedthroughfirstorder? . The −λ21Ec[n1/λ1]+λ21EM(2P).
perturbationexpansionofthefictitiousλ -interactingen-
1
ergy through second order equals: As shown in Sec. IID, the DS2DH functional defined in
Eq. (34) can be connected with conventional double hy-
Eλ1 =(cid:104)Φλ1|Tˆ+λ Wˆ +Vˆλ1|Φλ1(cid:105)+λ2E(2) +..., (31) brids when neglecting both second-order corrections to
0 0 1 ee 0 0 1 MP
the density as well as the density scaling in the correla-
where E(2) is the conventional MP2 energy correction tion functional.
MP
calculated with the λ -interacting orbitals and orbital
1
energies obtained from Eq. (30). Using the simplified
perturbation expansion for the second term in the right- D. Connection with conventional double hybrids
hand side of Eq. (29):
In order to connect regular double hybrids with the
(cid:104)Ψλ1|Wˆ |Ψλ1(cid:105)=(cid:104)Φλ1|Wˆ |Φλ1(cid:105)+2λ E(2) +..., one derived in Sec. IIC, we neglect both second-order
0 ee 0 0 ee 0 1 MP (32)
corrections to the density as well as the density scaling
5
in the correlation functional:
Let us now focus on the calculation of the orbitals. As
δn(2)(r)≈0, E [n ]≈E [n]. (36)
c 1/λ c pointed out in Sec. IIC, the DS2DH and thus the 2DH
exchange-correlation energies are based on HF-type or-
The DS2H and DS2DH exchange-correlation energies in
bitals calculated for a λ -interacting system. A natural
Eqs. (21) and (34) reduce then to two-parameter hybrid 1
choice of orbitals would therefore be the DS2-HF-OEP
(2H) and two-parameter double hybrid (2DH) exchange-
ones. Since density scaling is neglected, we will refer to
correlation energies, respectively:
themasλ -OEP-2Horbitalsandthecorresponding2DH
1
Eax,ac =a EHF[Φ]+(1−a )E [n] energywillbereferredtoasλ -OEP-2DH.Theλ -OEP-
xc,2H x x x x 1 1
(37) 2H scheme, which is an OEP-type calculation, can be
+(1−a )E [n], formulated as follows, according to Eqs. (13), (19) and
c c
(36) and (39):
and
Exacx,,2aDcH =axExHF[Φ]+(1−ax)Ex[n] Eλa1x-,OaEcP-2H =mvinEλa1x-,OaEcP-2H[v],
(38)
+(1−ac)Ec[n]+acEM(2P), Eλa1x-,OaEcP-2H[v]=(cid:110)(cid:104)Φλ1[v]|Tˆ+Vˆne|Φλ1[v](cid:105)
(43)
where we have introduced the two parameters a and a
x c
defined as follows: +EH[nΦλ1[v]]+axExHF[Φλ1[v]]
a =λ , (cid:111)
x 2
+(1−a )E [n ]+(1−a )E [n ] ,
x x Φλ1[v] c c Φλ1[v]
a =λ (2λ −λ ) (39)
c 1 2 1
where Φλ1[v] fulfills the λ1-interacting HF equation
=a2−(a −λ )2.
x x 1 (cid:16) (cid:17)
Tˆ+λ Uˆ [Φλ1[v]]+Vˆ |Φλ1[v](cid:105)=Eλ1[v]|Φλ1[v](cid:105),
Letusfirstnoticethat,bycontrasttoKSsecond-order 1 HF HF
perturbation theory? (KS-PT2), single excitations do (44)
notappearinthe2DHenergyexpression. Indeed,thelat- with Vˆ = (cid:82) drv(r)nˆ(r) and λ1 defined in Eq. (41). On
teroriginatesfromaMP-typecalculation,wheretheBril- the hand, for conventional double hybrids, the orbitals
louin theorem therefore applies, and which is based on a are obtained from a 2H calculation which can be formu-
partially λ -interacting system. In addition, it is readily lated as?
1
seen from Eq. (39) that, in this context, ac ≤ a2x. Inter- Eax,ac =min(cid:110)(cid:104)Φ|Tˆ+Vˆ |Φ(cid:105)+E [n ]+a EHF[Φ]
estingly, the various conventional double hybrids consid- 2H ne H Φ x x
Φ
ered in Table I fulfill this condition. Note that the one- (45)
parameterdoublehybrid(1DH)approximation? isrecov- (cid:111)
+(1−a )E [n ]+(1−a )E [n ] .
ered when λ = λ , or, in terms of a and a , a = a2. x x Φ c c Φ
1 2 x c c x
For given values of λ and λ , a unique set of a and a
1 2 x c TheminimizingdeterminantΦ˜ inEq.(45)fulfillsthe2H
parameters can be defined according to Eq. (39). Recip-
equation
rocally, for given a and a values, we can define the two
x c
scaling factors λ1 and λ2 as follows: (cid:16)Tˆ+Vˆ +a Uˆ [Φ˜]+(1−a )Vˆ [n ]
ne x HF x Hx Φ˜
λ =a ,
2 x (46)
(40) (cid:17)
(cid:112) +(1−a )Vˆ [n ] |Φ˜(cid:105)=E |Φ˜(cid:105),
λ =a ± a2−a . c c Φ˜ 2H
1 x x c
Whena =0,thecorrelationenergyisfullydescribedby and,therefore,canbeinterpretedastheHFdeterminant
c
thecorrelationfunctionalwhichmeansthatthefictitious associated to a λ -interacting system since, according to
2
λ -interactingsystemshouldbethenon-interacting(KS) Eq. (40), λ = a . However, as shown in Sec. IIC, the
1 2 x
one, that is λ = 0. The standard exchange-only-type 2DH exchange-correlation functional in Eq. (38) can be
1
KS-OEP scheme with a as fraction of exact exchange is justified when it is based on HF-type orbitals associated
x
thus recovered. We therefore conclude to a λ -interacting Hamiltonian. This condition ensures
1
(cid:112) thatsingleexcitationsdonotcontributetotheexchange-
λ =a − a2−a . (41)
1 x x c correlation energy. In fact, the conventional calculation
of the orbitals is simply deduced from Eq. (43) when re-
Interestingly,thelinearlyscaledone-parameterdoublehy-
placing the minimization over local potentials by a mini-
brid functional (LS1DH) derived recently by Toulouse et
al.? ,whichischaracterizedbya =a3,isrecoveredhere, mizationoversingledeterminants,whichisjustifiedonly
c x whenλ =λ thatisequivalenttoa =a2 (seeSec.IIB).
though density scaling is neglected, when λ1 equals 2 1 c x
As shown in Table I, conventional double hybrids do not
√
λLS1DH =a (1− 1−a ). (42) fulfillthelattercondition. Inthisrespect,doublehybrids
1 x x
6
based on the λ -OEP-2H approximation have a better and
1
justification than the regular ones. Let us finally con-
sider a possible alternative to the λ1-OEP-2H scheme, Exacx,,λa1c-2DH =axExHF[Φ˜(cid:48)]+(1−ax)Ex[n˜(cid:48)]
which would not require the calculation of an OEP and (52)
would have a computational cost similar to conventional +(1−a )E [n˜(cid:48)]+a E(2)(cid:48).
c c c MP
2H calculations. As shown in Sec. IIA, in the exact the-
oryandforafixedλ1 value, theOEPshouldnotdepend Let us stress that the λ1-2H and λ1-2DH energies as de-
on λ . As an approximation, referred to as λ -2H in fined in Eq. (50), and thus the corresponding exchange-
2 1
the following, we assume that this statement still holds correlation energies in Eqs. (51) and (52), are obtained
within the approximate λ -OEP-2H scheme. As a result, from the same energy expressions as in the regular 2H
1
anapproximatepotentialcanbeobtainedwhenchoosing, and 2DH schemes, using standard ax and ac values. The
inEq.(19),λ =λ insteadofλ =a . Inthisparticular difference comes from the orbitals which are calculated
2 1 2 x
case, according to Eqs. (22), (24), and (36), the λ -2H with the modified a(cid:48) and a(cid:48) coefficients.
1 x c
determinant Φ˜(cid:48) fulfills the modified 2H equation
(cid:16)
Tˆ+Vˆ +a(cid:48)Uˆ [Φ˜(cid:48)]+(1−a(cid:48))Vˆ [n ] E. Summary
ne x HF x Hx Φ˜(cid:48)
(47)
(cid:17) Atwo-parameterextensionoftheDS1DHschemepro-
+(1−a(cid:48))Vˆ [n ] |Φ˜(cid:48)(cid:105)=E(cid:48) |Φ˜(cid:48)(cid:105),
c c Φ˜(cid:48) 2H posed recently by Sharkas et al.? has been derived. It is
based on the explicit treatment of a fraction of mdEXX
where the regular a and a parameters have been re-
x c which requires, in the general case, the calculation of an
placed by λ and λ2, respectively:
1 1 OEP for a partially-interacting system. Computing this
(cid:112) OEP at the HF level of approximation leads to the DS2-
a →a(cid:48) =λ =a − a2−a
x x 1 x x c HF-OEP scheme where the energy has a DS2H form.
(48) In addition, it was shown that, using any approximate
(cid:16) (cid:112) (cid:17)
a →a(cid:48) =λ2 =2a a − a2−a −a . potential, DS2DHs can be defined. The connection be-
c c 1 x x x c c
tweenthelatterandregulardoublehybridsismadewhen
We thus ensure that the orbitals are computed from a neglecting both second-order corrections to the density
λ -interacting system. This procedure basically consists and density scaling. In this case, the DS2-HF-OEP ap-
1
in approximating the exact potential vλ1 by proximation reduces to the λ1-OEP-2H one and the cor-
responding DS2DH scheme (which is based on the λ -
1
v (r)=v (r)+(1−a(cid:48))δEHx[n ] OEP-2H potential) is then referred to as λ1-OEP-2DH.
λ1-2H ne x δn(r) Φ˜(cid:48) As an alternative to the λ1-OEP-2H calculation, the λ1-
(49) 2H scheme, where the OEP calculation is replaced by
+(1−a(cid:48)) δEc [n ]. a 2H one with modified exchange and correlation coef-
c δn(r) Φ˜(cid:48) ficients, has been proposed. A λ1-2DH approximation
could thus be defined from the λ -OEP-2DH energy ex-
1
UsingthispotentialandΦ˜(cid:48)asv0λ1 andΦλ01 inEq.(33),we pression using λ1-2H orbitals instead of the λ1-OEP-2H
obtain from Eqs. (36) and (39) the following expressions ones. In this work, only λ1-2H and λ1-2DH results will
for the λ -2DH energy: be shown. The OEP-based schemes, which are currently
1
under implementation, will be presented in a separate
Eax,ac =(cid:104)Φ˜(cid:48)|Tˆ+Vˆ |Φ˜(cid:48)(cid:105)+E [n˜(cid:48)]+a EHF[Φ˜(cid:48)] paper.
λ1-2DH ne H x x
+(1−a )E [n˜(cid:48)]
x x
(50) III. COMPUTATIONAL DETAILS
+(1−a )E [n˜(cid:48)]+a E(2)(cid:48)
c c c MP
Bothλ -2Handλ -2DHenergyexpressionsinEq.(50)
1 1
=Eax,ac +a E(2)(cid:48), can be simply implemented using a regular DFT code
λ1-2H c MP
that can perform double hybrid calculations. A develop-
where n˜(cid:48) denotes the density of Φ˜(cid:48) and E(2)(cid:48) is the regu- ment version? of the DALTON program package? has
larMP2energycorrectioncalculatedwithMλP-2Horbitals beenusedinthiswork. Inafirststep,aλ1-2Hcalculation
1
and orbital energies. The λ -2H and λ -2DH exchange- is done. It consists in performing a 2H calculation with
1 1
themodifiedexchangeandcorrelationcoefficientsdefined
correlation energies are therefore expressed respectively
inEq.(48). Thecorrespondingorbitalsandorbitalener-
as
giesarethenused,inasecondstep,tocomputethescaled
Exacx,,λa1c-2H =axExHF[Φ˜(cid:48)]+(1−ax)Ex[n˜(cid:48)] MP2 term. In a third step, the λ1-2H energy is obtained
(51) from a regular one-iteration 2H calculation, using the
+(1−ac)Ec[n˜(cid:48)], λ1-2H orbitals as starting orbitals. Three conventional
7
double hybrids have been considered: the B2-PLYP? etal.? study,resultsareexpectedtochangesignificantly.
and B2GP-PLYP? functionals based on the Becke 88
(B) exchange functional and the Lee-Yang-Parr (LYP)
correlation functional, and the PBE0-DH? functional
which is based on the Perdew-Burke-Ernzerhof (PBE) V. CONCLUSIONS
exchange functional? and the Perdew-Wang (PW) cor-
relation functional? . The corresponding exchange and Arigorousderivationoftwo-parameterdoublehybrids
correlation coefficients are given in Table I. Calculations (2DHs) has been presented. It is based on the combi-
have been performed on a test set consisting of H2, N2, nation of the DS1DH scheme of Sharkas et al.? with
Be2, Mg2 and Ar2. The following basis sets were used: the explicit treatment of a fraction of multideterminan-
cc-pVQZ? for H2 and N2, aug-cc-pVQZ? for Be2 and tal exact exchange. The connection with regular double
Mg2, and aug-cc-pVTZ? for Ar2. hybridsismadewhenneglectingbothdensityscalingand
second-order corrections to the density. It then appears,
in this context, that the fraction of second-order Møller-
IV. RESULTS AND DISCUSSION Plesset (MP2) energy correlation is smaller or equal to
thesquareofthefractionofHartreeFock(HF)exchange.
In this section we compare, in terms of accuracy, the Interestingly, various conventional semi-empirical double
regular B2-PLYP, B2GP-PLYP and PBE0-DH double hybridsfulfillthiscondition. Inthelightofthosederiva-
hybrids with their λ variants, as defined in Sec. IID. tions,anewprocedureforcalculatingtheorbitals,which
1
ThepotentialcurvescomputedforH andN (seeFig.1) is more justified than the one used routinely, has been
2 2
show a systematic lowering of the total energy when proposed. Itstillconsistsinperformingatwo-parameter
using λ -double hybrids instead of the regular ones. hybrid calculation, but with modified exchange and cor-
1
This can be analyzed when decomposing the difference relationcoefficients. Preliminaryresultspresentedinthis
between the λ -2DH and regular double hybrid energies work show that, in such a scheme which is referred to
1
i(nsetoe aEqssi.ng(l4e5)hyabnridd (t5e0rm)) ∆anEdSHa =scaEleλad1x-,2aHMc P−2Et2aeHxr,amc avasluλe1,-2lDarHge,rththeaMnPth2eerneegruglyarcoonnter.ibAutsioanriess,uilnt,aλb1s-o2lDutHe
a ∆E = a (E(2)(cid:48) −E(2)) (see Eq. (50)). As illus- and regular double hybrid potential curves can, in some
c MP2 c MP MP cases, differ significantly. In particular, for the tested
trated in Fig. 2 (a), for H , the λ -2H energy is always
2 1 weakly bound dimers, the λ variants bind systemati-
greater than the regular 2H one (∆E ≥ 0) which is 1
SH
cally more than the regular ones, which is often but not
duetothefactthattheλ -2Horbitalsarenotoptimized
1
always an improvement. Including density scaling in the
for the regular 2H energy expression (based on a and
x
correlation functionals may of course change the results
a ) but for a modified one (based on a(cid:48) and a(cid:48)). This
c x c significantly. This still needs to be investigated. More-
positive difference is then compensated when adding the
over, optimized effective potentials (OEPs) based on a
scaled MP2 contribution. According to Eq. (48), the
partially-interacting system could also be used to gen-
fraction a(cid:48) of HF exchange used to compute the λ -2H
x 1 erate proper orbitals . Work is currently in progress in
orbitals is lower than the regular fraction a . This leads
x
those directions.
to a smaller HOMO-LUMO gap (see Fig. 2 (b)) and
thus to a larger scaled MP2 correction. Note, however,
that B2GP-PLYP gives the largest a ∆E term in
c MP2 Acknowledgments
absolute value while PBE0-DH, for which a(cid:48) differs the
x
most from a , has the largest reduction (in absolute
x
E.F.thanksANR(contractDYQUMA),AndrewTeale
value) of HOMO-LUMO gap. It could be explained
for his help in computing the λ -2DH energy and his
by the fact that the scaled MP2 correction is sensitive 1
commentsonthiswork,aswellasAlexandrinaStoyanova
not only to the orbital energies but also to the orbitals.
and Julien Toulouse for fruitful discussions.
Results obtained for N (not shown) lead to the same
2
conclusion. Returning to the potential curves in Fig. 1,
we finally note that the λ variant slightly improves the
1
accuracy of the double hybrids in the vicinity of the
equilibrium distance.
Concerning the weakly-bound systems Be , Mg and
2 2
Ar , the interaction energy curves in Fig. 3 show that
2
regular double hybrids and their λ variants behave
1
similarlyatlargedistancesbutthelatterbindmorethan
the former. This often leads to slightly more accurate
results unless the regular double hybrid, like PBE0-DH
for Be and Mg , already overbinds. It would of course
2 2
beinterestingto evaluate theeffects ofdensityscaling in
thecorrelationfunctional. Indeed,inthelightofSharkas
8
FIGURE CAPTIONS
Figure 1: Potential curves for H (top) and N (bot-
2 2
tom). The ”exact” curves are taken from Ref.?
Figure 2: Single hybrid (∆E ) and scaled MP2
SH
(a ∆E )energydifferencesbetweenλ -andcon-
c MP2 1
ventional B2-PLYP, B2GP-PLYP and PBE0-DH
doublehybridscalculatedforH withrespecttothe
2
bond distance (left); HOMO-LUMO gap for both
λ - and conventional B2-PLYP, B2GP-PLYP and
1
PBE0-DH double hybrids calculated for H with
2
respect to the bond distance (right).
Figure 3: Interaction energy curves for Be , Mg and
2 2
Ar . Theaccurate? ,experimental? andCCSD(T)
2
curves are taken as reference.
6
9
FIG. 1: Fromager, Journal of Chemical Physics
FIG. 1: Fromager, Journal of Chemical Physics
H (cc−pVQZ)
2
−1.158 B2−PLYP
λ −B2−PLYP
1
−1.16 B2GP−PLYP
λ −B2GP−PLYP
1
−1.162 PBE0−DH
λ −PBE0−DH
) 1
u. −1.164 exact
.
a
(
y −1.166
g
r
e
n −1.168
e
−1.17
−1.172
−1.174
0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84
bond distance (Å)
N (cc−pVQZ)
2
B2−PLYP
−109.1 λ −B2−PLYP
1
B2GP−PLYP
λ −B2GP−PLYP
1
PBE0−DH
−109.2
λ −PBE0−DH
) 1
u. exact
.
a
(
y −109.3
g
r
e
n
e
−109.4
−109.5
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
bond distance (Å)
10
FIG. 2: Fromager, Journal of Chemical Physics
7
FIG. 2: Fromager et al, Journal of Chemical Physics
(a) H (cc−pVQZ) (b) H (cc−pVQZ)
2 2
1.5 0.8
B2−PLYP (ΔESH) B2−PLYP
B2GP−PLYP (ΔESH) 0.75 λ1−B2−PLYP
ence (mH) 0 .015 B2GPBBP2E−−P0PPB−LLEDYY0HPP− D(((aaaHccc ΔΔΔ(ΔEEEEMMMSPPPH222)))) O gap (a.u.) 0 .06.57 λ1−λBB1−22GGPPBBPPEE−−00PP−−LLDDYYHHPP
er M 0.6
gy diff −0.5 O−LU 0.55
ener −1 HOM 0.5
0.45
−1.5
0.4
0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84
bond distance (Å) bond distance (Å)