Range-dependent Adiabatic Connections
A. M. Teale
∗, S. Coriani
†,∗and T. Helgaker
∗∗Department of Chemistry, Centre for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway
†Dipartimento di Scienze Chimiche, Università degli Studi di Trieste, Via Licio Giorgieri 1, I-34127 Trieste, Italy
Abstract. Recently, we have implemented a scheme for the calculation of the adiabatic connection linking the Kohn–Sham system to the physical, interacting system. This scheme uses a generalized Lieb functional, in which the electronic-interaction strength is varied in a simple linear fashion, keeping the potential or the density fixed in the process. In the present work, we generalize this scheme further to accommodate arbitrary two-electron operators, allowing the calculation of adiabatic connections following alternative paths as outlined by Yang [J. Chem. Phys.109, 10107 (1998)]. Specifically, we examine the error-function and Gaussian-attenuated error-function adiabatic connections. We explore the high-density and strong static- correlation regimes for two-electron systems. The resulting adiabatic connections give an alternative view of the exchange–
correlation problem and their utility for the development of new exchange–correlation functionals in Kohn–Sham and range- separated hybrid schemes is discussed.
Keywords:Density Functional Theory, Adiabatic Connection, Exchange–Correlation, Lieb Functional, FCI
INTRODUCTION
The adiabatic-connection (AC) formula for the exchange–correlation energy [1, 2, 3, 4, 5] in density-functional theory (DFT) has motivated the construction of orbital-dependent functionals [6, 7], which represent some of the most successful approximations in widespread use. The AC formula arises from a consideration of the link between the Kohn–Sham noninteracting system and the physical, interacting system as a function of the interaction strength. A number of studies have examined the AC using approximate methods [8, 9, 10, 11, 12, 13, 14, 15, 16], and some high-accuracy studies have been carried out for few-electron atomic systems [17, 18, 19, 20, 21, 22]. Recently, we presented an implementation of a scheme for the calculation of accurate AC curves fromab initiodensities [22, 23]
via optimization of the Lieb functional [24]. Our implementation considers not only the usual density-fixed AC, of relevance in DFT, but also the potential-fixed AC, of relevance in potential functional theory (PFT), in which the the fully interacting system is related to the noninteracting, bare-nucleus system [22] (with the potential fixed at the external potential from the nuclei). The same connection was considered independently by Gross and Proetto [25].
In the present work, we focus on the density-fixed adiabatic connection for general integration paths; results for the potential-fixed connection will be presented in a forthcoming publication.
Most previous studies of the AC consider only the case in which the electron–electron repulsion is modulated in a simple linear fashion, by introducing a scaling of the two-electron interaction. However, as was pointed out by Yang [26], this choice is not unique. In fact, the electronic interaction may be modified by any function that smoothly connects the noninteracting and physical systems. These generalized ACs are of particular relevance to hybrid theories, constructed to combine Kohn–Sham DFT and wave-function approaches as proposed by Savin [27].
With an appropriate modification of the electronic interaction, it is possible to attempt the construction of hybrid theories, in which short-range interactions are treated by DFT and long-range interactions by a suitable choice of wave-function methodology. Recently, a variety of short-range DFT functionals have been developed within the local-density approximation (LDA) [27, 28], the generalized gradient approximation (GGA) [29, 30, 31] and the meta-GGA [32]. Several implementations of these hybrid schemes exist combining short-range DFT functionals with long-range Hartree–Fock (HF) [33], configuration-interaction (CI) [34, 35], second-order Møller–Plesset (MP2) [33], coupled-cluster (CC) [30], multiconfigurational self-consistent field (MCSCF) [36, 37] andn-electron valence second- order perturbation-theory (NEVPT2) [38] methods. We also note that range separation of only the exchange interaction has been explored in the context of developing new DFT exchange–correlation functionals. Notable examples are the long-range corrected (LC) functionals developed by Hirao and co-workers [39, 40], theω-PBEh and HSE functionals developed by Scuseria and co-workers [41, 42] and the CAM-B3LYP functional developed by Handy and co- workers [43, 44]. These functionals emphasize either short-range [41, 42] or long-range [39, 40, 43] interactions;
a variant emphasizing the middle range has also been reported [45]. Finally, in a different context, we note the use of a family of similar interactions by Gill and co-workers to remove the long-range tails of the Coulomb interaction [46, 47, 48].
Whilst some studies of nonlinear ACs have been carried out from the point of view of calculating short-range DFT exchange–correlation energies and potentials [21, 49], no direct studies of the generalized, range-dependent AC integrand have been presented, in contrast to the linear case. Given the central role that the AC formulation plays in the theory underlying range-separated approaches, we here generalize our implementation of an optimization scheme for the Lieb functional [22] to electronic interactions weighted by the error function and the Gaussian-attenuated error function. We commence by introducing the theory of ACs with general two-electron operators and then briefly review our approach to optimization of the Lieb functionals and calculation of the AC integrands. Here we focus on details specific to this generalized scheme, referring the reader to Refs. [22] and [23] for details of the optimization scheme.
Next, we present results for the calculation of ACs corresponding to FCI densities for some simple two-electron systems. In the final section, we make some concluding remarks and discuss directions for future work.
THEORY Consider anN-electron system described by the Hamiltonian
Hˆλ[v] =Tˆ+Wˆλ+
∑
i
v(ri), 0≤λ≤1 (1)
wherev(r)is the external potential atr, ˆT is the kinetic-energy operator Tˆ=−1
2
∑
i
∇2i (2)
and ˆWλ is a generalized electron-interaction operator depending on a coupling-strength parameter λ that varies betweenλ=0 (the noninteracting system) andλ=1 (the fully interacting system),
Wˆλ=1 2
∑
i$=j
wλ(ri j), w0(ri j) =0, w1(ri j) =1/ri j (3) We now introduce the ground-state energyEλ[v]as a functional of the external potential and the energyFλ[ρ]as a functional of the electron density by the following constrained minimizations [24, 50, 51, 52] over density matrices ˆγ:
Eλ[v] = inf
γ→Nˆ Tr ˆHλ[v]γˆ=Tr ˆHλ[v]ˆγλv (4) Fλ[ρ] =inf
γ→ρˆ Tr ˆHλ[0]ˆγ=Tr ˆHλ[0]γˆλρ (5) where we denote the minimizers by ˆγλvand ˆγλρ, respectively. Whereas a minimizer ˆγλρalways exists in Eq. (5), this is not so for the minimization in Eq. (4), where ˆγλv only exists for those potentialsvthat support an electronic ground state for a given interaction strengthλ. In the following, we shall assume that a minimizer exists.
As first discussed by Lieb [24], the ground-state energy as a functional of the external potentialEλ[v]and the energy as a functional of the densityFλ[ρ]are convex conjugate functionals (mutual Legendre–Fenchel transforms):
Eλ[v] = inf
ρ∈X(Fλ[ρ] + (v|ρ)) (6)
Fλ[ρ] =sup
v∈X∗
(Eλ[v]−(v|ρ)) (7)
where the domainsX andX∗are reflexive Banach spaces such that(v|ρ) =!v(r)ρ(r)dris finite for allρ∈X and v∈X∗. In general, we obtain from Eqs. (6) and (7) the Fenchel inequality
Eλ[v]≤Fλ[ρ] + (v|ρ) (8)
which holds for allvandρ. The conditions for a minimizing densityρin Eq. (6) and for a maximizing potentialvin Eq. (7) are equivalent and may be expressed in the following manner:
Eλ[v] =Fλ[ρ] + (v|ρ) ⇔ δEλ[v]
δv(r) =ρ(r) ⇔ δFλ[ρ]
δ ρ(r) =−v(r) (9)
An external potentialvand a densityρthat together satisfy Eq. (9) are said to be conjugate. For a given potentialv, one or more conjugate densitiesρ may be found provided the potential supports a (possibly degenerate)N-electron ground state. Conversely, anN-electron densityρhas a conjugate potentialv(unique to within an additive constant) providedρisv-representable.
We now introduce the AC by relating the functionalFλ[ρ]forλ>0 to the corresponding noninteracting quantity F0[ρ]. Denoting differentiation with respect toλby prime and using the Hellmann–Feynman theorem, we obtain
Fλ[ρ] =F0[ρ] +
"λ
0 Fλ([ρ]dλ=Ts[ρ] +
"λ
0
Wλ[ρ]dλ (10) where we have introduced the noninteracting kinetic-energy functional Ts[ρ]and the AC integrandWλ[ρ]as the expectation value of the differentiated two-electron operator ˆWλ( with respect to the density matrix ˆγλρ optimized at interaction strengthλfrom Eq. (5):
Ts[ρ] =Tr ˆH0[0]γˆ0ρ (11)
Wλ[ρ] =Tr ˆWλ(γˆλρ (12) It is customary to decompose the total interaction energy in Eq. (10) in the manner
"λ
0
Wλ[ρ]dλ=Jλ[ρ] +Ex,λ[ρ] +Ec,λ[ρ] (13)
where the classical Coulomb functionalJλ[ρ], the exchange functionalEx,λ[ρ], and the correlation functionalEc,λ[ρ]
are given by
Jλ[ρ] =1 2
" "
wλ(r12)ρ(r1)ρ(r2)dr1dr2 (14)
Ex,λ[ρ] =Tr ˆWλγˆ0ρ−Jλ[ρ] (15)
Ec,λ[ρ] =
"λ
0
Wc,λ[ρ]dλ, Wc,λ[ρ] =Tr ˆWλ(#γˆλρ−γˆ0ρ$
(16) The exchange and correlation energies may be combined to give the exchange–correlation energy, which by combina- tion of Eq. (15) with Eq. (16) is given by
Exc,λ[ρ] =
"λ
0
Wxc,λ[ρ]dλ, Wxc,λ[ρ] =Tr ˆWλ(γˆλρ−Jλ([ρ] (17) In the following, we shall study the AC for the full electronic interaction energy of the helium isoelectronic series and of the H2molecule at different internuclear separations, with different choices of ˆWλ. The decomposition of these density-fixed ACs into exchange–correlation and correlation-only contributions will be reported in a forthcoming publication along with the corresponding quantities for the potential-fixed AC.
Yang [26] observed that, since the AC integrand is determined entirely by the functional values at the end points of the integration (λ2>λ1)
"λ2
λ1
Wλ[ρ]dλ=Fλ2[ρ]−Fλ1[ρ] (18)
we may choose ˆWλ freely in Eq. (1) provided its end-point values (typically 0 and 1) are unaffected. This idea has been used by Savin and co-workers to justify the construction of a variety of hybrid theories that merge wave-function approaches with DFT, see for example Refs. [27, 34, 21, 53, 54, 49]. Whilst some studies have appeared examining
1 2 3 4 4
8
ws
1 2 3 4
4 8
we
1 2 3 4
4 8
wg
1 2 3 4
4 8
dws!dΛ
1 2 3 4
4 8
dwe!dΛ
1 2 3 4
4 8
dwg!dΛ
FIGURE 1. Attenuated operators (top row) and theirλderivatives (bottom row) as functions ofr12forλ=0 (pink line),λ=1/4 (blue line),λ=1/2 (green line),λ=3/4 (red line) andλ=1 (black line).
the integrated quantitiesExc[ρ]andEc[ρ]for the density-fixed connection [21, 49], no explicit study of the integrands involved, varying the path between the noninteracting and interacting systems, has been carried out. In the present work, we consider the following general forms forwλ(ri j)in Eq. (3):
wsλ(ri j) = λ ri j
(standard) (19)
weλ(ri j) =erf%
1−λλ ri j
&
ri j (error-function) (20)
wgλ(ri j) = erf%
1−λλ ri j
&
ri j − 2
√π
' λ
1−λ (
exp )
−1 3
' λ
1−λ (2
r2i j
*
(Gaussian-attenuated error-function) (21) The choicewsλin Eq. (19) represents the standard range-independent AC, depending linearly onλ. Asλincreases, the interaction is turned on uniformly for all interelectronic separationsri j. By contrast, with the error-function connection weλin Eq. (20) and Gaussian-attenuated error-function connectionwgλin Eq. (21), the interaction is turned on in a range- dependent, nonuniform manner by the functions erf(µri j)and exp(−µ2r2i j/3)ofri j, whereµ=λ/(1−λ)varies over the range 0≤µ≤∞whenλ increases from 0 to 1. As a result, with these two connections, long-range interactions are accounted for first and short-range interactions last. To illustrate the difference between the above connections, we have in Figure 1 plotted the functions in Eqs. (19)–(21) and their derivatives as functions ofri j, for four different values ofλ. Theλ dependence of the derivatives is relevant since, in the evaluation of the AC integrand in Eq. (12), we calculate the expectation value of the density matrix with these derivatives.
RESULTS
The helium isoelectronic series has been extensively studied and poses a significant challenge for approximate exchange–correlation functionals, particularly asZ increases [17, 18, 55, 56, 57, 58, 59]. In the present work, we examine the systems with 1≤Z≤10 using the uncontracted aug-cc-pVQZ basis set [60, 61, 62], noting that uncontraction is essential to describe the compact densities accurately. The total energy and its components are listed in Table 1.
In Figure 2, we have plottedWλ[ρ]againstλ for 1≤Z≤10 wherewsλ is in Pane (a),weλ in Pane (b), andwgλ in Pane (c). Comparing the AC curves arising from the different choices ofwλ in this figure, we recall that each curve
TABLE 1. Energy components of the helium isoelectronic series in the uncontracted aug-cc- pVQZ basis (atomic units)
Z Etot(Z) Ts[ρ] (v|ρ) !Wλ[ρ]dλ J[ρ] Ex[ρ] Ec[ρ] Tc[ρ] 1 −0.527 0.502 −1.374 0.345 0.773 −0.386 −0.041 0.028 2 −2.903 2.865 −6.751 0.983 2.048 −1.024 −0.041 0.036 3 −7.275 7.235 −16.122 1.612 3.300 −1.650 −0.038 0.036 4 −13.650 13.610 −29.497 2.237 4.552 −2.276 −0.039 0.037 5 −22.025 21.984 −46.872 2.862 5.803 −2.901 −0.039 0.038 6 −32.400 32.359 −68.247 3.488 7.054 −3.527 −0.039 0.038 7 −44.775 44.734 −93.622 4.113 8.304 −4.152 −0.039 0.038 8 −59.150 59.108 −122.996 4.738 9.554 −4.777 −0.039 0.039 9 −75.525 75.483 −156.371 5.363 10.804 −5.402 −0.039 0.039 10 −93.900 93.857 −193.746 5.988 12.055 −6.027 −0.040 0.039
FIGURE 2. AC curves (atomic units)Wλ[ρ]for the helium isoelectronic series with 1≤Z≤10 forwsλ in Pane (a), forweλin Pane (b) and forwgλin Pane (c). In all panes, the curves increase with increasingZ.
represents the expectation value of ˆWλ(with ˆγλρ, optimized with the two-electron operator ˆWλ. The standard connection wsλin Pane (a) yields nearly straight lines, representing a situation where the interactions are turned on uniformly for all interelectronic separations. Theweλcurves in Pane (b) give the same total interactions as those in Pane (a) but have very different shapes, since the interactions are now first turned on for large interelectronic separations and subsequently for short separations. The AC curves are therefore no longer linear but contain a peak at that value ofλwhere most of the interactions are recovered. ForZ=1, the peak is broad and occurs already atλ≈0.1, reflecting the large range of interelectronic separations that contribute to the interactions in this diffuse system. For the most compact system with Z=10, there is a sharp peak atλ≈0.87, indicating that most interactions occur at about 0.2a0–0.3a0. Thewgλ plots in Pane (c) are similar to those in Pane (b) but have shaper peaks, reflecting the higher locality ofwg(λ, see Figure 1.
Having explored dynamic correlation in the high-density limit for two-electron atomic systems, we now explore the transition from dynamic correlation to static correlation by stretching the H2 molecule. The H2 molecule is a prototypical system that can be considered as representative for the dissociation of electron-pair bonds in general. The fully interacting wave functionΨ1is a singlet at all all geometries [63], consistent withρα(r) =ρβ(r) =ρ(r)/2, as is imposed in spin-restricted Kohn–Sham theory. In the present work, all calculations use the restricted formalism. For more detailed discussion, see Ref. [22]. We now examine the AC curves for the bond lengthsR=0.7, 1.4, 3.0, 5.0, 7.0 and 10.0 (in units ofa0) for the different choices of two-electron interaction in Eqs. (19)–(21). In all calculations, the aug-cc-pVQZ basis set [60, 61, 62] is used.
In Figure 3, we have plottedWs
λ[ρ]in Pane (a),We
λ[ρ]in Pane (b), andWg
λ[ρ]in Pane (c). For the short bond distancesR=0.7a0and 1.4a0, the shape of the AC curvesWs
λ[ρ]in Pane (a) are similar to those for helium in Pane (a) of Figure 2 and are indicative of the quadratic dependence of dynamical correlation energy onλ. However, as the bond is stretched, the curves in Pane (a) bend more sharply, the changes as a function ofλbecoming localized to the low-λ end of the curves. Except for very small coupling strengths, the curve forR=10a0is horizontal at 0.100Eh, reflecting the complete absence of dynamical correlation in this system.
Turning our attention to the range-separated AC curves in Panes (b) and (c) of Figure 3, we note how long- range interactions become more dominant with increasing separationR. This behaviour is particularly pronounced
TABLE 2. Energy components of the H2molecule in the aug-cc-pVQZ basis (atomic units)
R Etot(R) Ts[ρ] (v|ρ) !Wλ[ρ]dλ J[ρ] Ex[ρ] Ec[ρ] Tc[ρ]
0.7 −0.921 1.731 −4.869 0.788 1.653 −0.827 −0.039 0.033 1.4 −1.174 1.141 −3.650 0.621 1.323 −0.661 −0.041 0.033 3.0 −1.057 0.828 −2.619 0.400 0.955 −0.477 −0.077 0.042 5.0 −1.004 0.953 −2.382 0.226 0.820 −0.410 −0.184 0.022 7.0 −1.000 0.993 −2.284 0.147 0.767 −0.384 −0.236 0.005 10.0 −1.000 1.000 −2.199 0.100 0.725 −0.362 −0.262 0.000
FIGURE 3. AC curves (atomic units)Wλ[ρ]for H2withR=0.7a0,1.4a0,3.0a0,5.0a0,7.0a0and 10.0a0forwsλin Pane (a), for weλin Pane (b) and forwgλin Pane (c). In all panes, the curves fall with increasingRatλ=0.8.
for the Gaussian-attenuated error-function curves, which develop a semi-discontinuity atλ≈0.43 forR=10.0a0. For λ>0.43, thewgλ operator only samples interactions between electrons located less than about 8a0apart. Since there are few such interactions in a system consisting of two hydrogen atoms 10a0apart, theWλ[ρ]curve drops to zero aroundλ=0.43.
CONCLUSIONS
We have examined the AC for generalized, range-dependent two-electron interactions by studying the helium iso- electronic series and the stretching of the hydrogen molecule. These prototypical systems exhibit a range of densities (diffuse and compact) and types of correlation (dynamic and static). Standard, range-independent AC curves were compared with range-dependent curves obtained by attenuating the two-electron interaction with the error function and with a Gaussian-attenuated error function. For the helium isoelectronic series, the range-dependent ACs displayed a peak that moved to large values of the interaction strengthλas the density became more compact, these peaks being most compact for the Gaussian-attenuated error function. For H2, this peak moved to smallerλ values as the bond was stretched and static correlation began to dominate, the striking feature of the Gaussian-attenuated curves being the development of a semi-discontinuity for large bond distancesRwhere the AC became essentially zero beyond a given coupling strength. This corresponds to the lack of interactions at a range below approximately 8.0a0, consistent with the stretched H2molecule.
In the present work, we have utilized the error-function-based operators to modify the total electronic interaction.
The extent to which this approach is helpful for the development of exchange–correlation functionals for practical use is being investigated. However, the ability to relate the features of these integrands to features of the electronic densi- ties, in addition to their end points corresponding to known constants, would seem to offer some promise. Future work will include the investigation of potential-fixed ACs for these modified paths along with the decomposition of the total AC into its exchange–correlation and correlation only contributions. The alternative view of the exchange–correlation problem offered by these ACs may prove useful for the development of new exchange–correlation functionals for use in the Kohn–Sham scheme. In addition, the utility of these connections in the context of range-separated schemes will be explored.
ACKNOWLEDGMENTS
This work has been supported by the CoE Centre for Theoretical and Computational Chemistry through Grant No.
179568/V30. A.M.T. gratefully acknowledges support from the Norwegian research council through Grant No. 171185 and the hospitality of Università degli Studi di Trieste during a visit for part of this work.
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