Range separation combined with the Overhauser model: Application to the H molecule along the dissociation curve
The combination of density-functional theory with other approaches to the many-electron problem through the separation of the electron-electron interaction into a short-range and a long-range contribution (range separation) is a successful strategy, which is raising more and more interest in recent years. We focus here on a range-separated method in which only the short-range correlation energy needs to be approximated, and we model it within the “extended Overhauser approach”. We consider the paradigmatic case of the H molecule along the dissociation curve, finding encouraging results. By means of very accurate variational wavefunctions, we also study how the effective electron-electron interaction appearing in the Overhauser model should be in order to yield the exact correlation energy for standard Kohn-Sham density functional theory.
Kohn-Sham (KS) density functional theory (DFT) (see, e.g., Kohn (1999)) is a successful method for electronic structure calculations, thanks to its unique combination of low computational cost and reasonable accuracy. In the Kohn-Sham formalism, the total energy of a many-electron system in the external potential is rewritten as a functional of the one-electron density ,
In Eq. (1), is the kinetic energy of a non-interacting system of fermions (usually called KS system) having the same one-electron density of the physical, interacting, system. The Hartree energy is the classical repulsion energy, , and the exchange-correlation functional must be approximated. Minimization of Eq. (1) with respect to the spin-orbitals forming the KS determinant lead to the KS equations. Thus, instead of the physical problem, in KS DFT we solve the hamiltonian of a model system of non-interacting fermions, and we recover the energy of the physical system via an approximate functional.
Despite its success in scientific areas ranging from material science to biology, approximate KS DFT is far from being perfect, and many fundamental issues still need to be addressed. In particular, KS DFT encounters difficulties in handling near-degeneracy correlation effects (rearrangement of electrons within partially filled shells), and in taking into account long-range van der Waals interaction energies (crucial, e.g., for layered materials and biomolecules). In principle, all the shortcomings of KS DFT come from our lack of knowledge of the exchange-correlation functional, and a huge effort is put nowadays in trying to improve the approximations for (for recent reviews see, e.g., Mattsson (2002); Perdew et al. (2005)).
An alternative strategy to overcome the problems of DFT is range separation: the electron-electron interaction is split into a long-range and a short range part, and the two are treated at different levels of approximation Stoll and Savin (1985); Savin (1996); Leininger et al. (1997); Iikura et al. (2001); Kamiya et al. (2002); Y. Tawada et al. (2004); Pollet et al. (2002); Toulouse et al. (2004); Ángyán et al. (2005); Toulouse et al. (2005a); Goll et al. (2005, 2006); Gerber et al. (2007); Fromager et al. (2007); Fromager and Jensen (2008); Heyd et al. (2003); Heyd and Scuseria (2004a); Heyd and Scuseria (2004b); Vydrov et al. (2006); Vydrov and Scuseria (2006); Henderson et al. (2007); Baer and Neuhauser (2005); Baer et al. (2006); Livshits and Baer (2007). Prof. Hirao has been a pioneer in this field, investigating the effect of range separation on the exchange energy with remarkable success (see, e.g., Iikura et al. (2001); Kamiya et al. (2002); Y. Tawada et al. (2004)).
The variant of range separation that we consider here Stoll and Savin (1985); Savin (1996); Leininger et al. (1997); Pollet et al. (2002); Toulouse et al. (2004); Ángyán et al. (2005); Toulouse et al. (2005a); Goll et al. (2005, 2006); Gerber et al. (2007); Fromager et al. (2007); Fromager and Jensen (2008) can be viewed as a way to remove the constraint that the model system be non-interacting: instead of the KS system, one can define a long-range-only-interacting system (whose wavefunction is thus multideterminantal) having the same density of the physical system. The remaining part of the energy is then approximated with a short-range exchange-correlation functional. The resulting long-range-only hamiltonian, being weakly interacting (and without the electron-electron cusp), can be treated at a reasonable computational cost with standard wavefunction methods: in general, the needed configuration space to achieve good accuracy is small, and often second-order perturbation theory suffices. At the same time, this long-range interaction, albeit small, can make the corresponding wavefunction capture near-degeneracy effects and long-range van der Waals energies. Provided that the energy functionals are correctly redefined, there is no double counting of the energy, and the method is in principle exact, as it is KS DFT.
As mentioned, this range-separated multideterminant DFT needs an approximation for the short-range exchange-correlation functional. One can follow the same path as for KS DFT: start with the local-spin-density approximation (LSDA), consistently constructed as the difference between the standard LSD functional and the exchange-correlation energy of an electron gas with long-range-only interaction Paziani et al. (2006), and then add gradient corrections (GGA) Goll et al. (2005); Toulouse et al. (2005b); Goll et al. (2006); Fromager et al. (2007), and eventually meta-gradient corrections (mGGA). However, this path, which proved highly successful for KS DFT, may not be the best for a scheme in which long-range correlations are explicitly taken into account by wavefunction methods. Indeed, in most cases there is no improvement when passing from LSDA to GGA Goll et al. (2005, 2006, 2007), with the exception of hydrogen-bonded complexes Goll et al. (2008).
In recent years we have extended the “Overhauser model”, an approximate method to calculate the short-range part of the pair density in the uniform electron gas, to systems of nonuniform density Gori-Giorgi and Savin (2005); Gori-Giorgi and Savin (2006a, 2007b, 2007a), finding that it yields an accurate description of the short-range part of the spherically- and system-averaged pair density (intracule density) of small atoms. In Ref. Gori-Giorgi and Savin (2007b) we have combined the Overhauser equations with the Kohn-Sham equations in a self-consistent way, recovering full CI total energies within 1 mH for the He isoelectronic series. In Ref. Gori-Giorgi and Savin (2007a) we have shown that, unlike all the available correlation functionals Katriel et al. (2006), the model works equally well for the high-density limit of the He and the Hooke’s atom series. Thus, on one hand, the Overhauser model seems to be a very good candidate to construct short-range correlation energy functionals. On the other hand, we have tested it only on systems dominated by dynamical correlation: in the He atom, the Overhauser model yields essentially the exact KS correlation energy. However, when we move to systems with strong static correlation we expect the Overhauser model to be unable to yield good results. The combination of the Overhauser model with range-separated multideterminant DFT seems then natural: it can be viewed as a way to produce an adapted short-range correlation functional for the range-separated multideterminant DFT, or as a way to add the description of static correlation to the Overhauser model.
In this work we combine the Overhauser model with range-separated multideterminant DFT, applying it to the case of the H molecule along the dissociation curve, thus analyzing also the case of strong static correlation, as the dissociation limit is approached. The paper is organized as follows. After briefly reviewing in Secs. II and III the basic equations of range-separated multideterminant DFT and of the extended Overhauser model, we first analyze in Sec. IV, using very accurate variational wavefunctions Rychlewski et al. (1994); Cencek et al. (1995); Cencek and Kutzelnigg (1996), how the “exact” electron-electron interaction which appears in the Overhauser model (and that it is usually approximated with a physically-motivated interaction) should be as the H molecule is stretched. This analysis shows the difficulty of modeling static correlation within the Overhauser model. Since the model is only able to describe correlation, we combine it with a generalized OEP scheme for multideterminant DFT, which is described in Sec. V. The combination of the two methods is then presented in Sec. VI, with results for the H molecule. The last Sec. VII is devoted to conclusions and perspectives.
Ii Multideterminant DFT via range separation
In Eq. (2) the expectation of the kinetic energy operator plus the Coulomb electron-electron repulsion operator is minimized over all wavefunctions yielding the density . The universality of the functional stems from the fact that and are the same for every electronic system of given particle number . Kohn and Sham Kohn and Sham (1965) introduced another functional, of Eq. (1), by replacing in Eq. (2) with zero,
and used for approximating an important part of . In Eq. (3), and in the rest of this paper, denotes a non-interacting wavefunction (thus in the majority of cases a single Slater determinant). Similarly, we can introduce a functional for a long-range-only interaction (here chosen using the error function, with the real parameter governing the cutoff of the short-range part),
In this way we have
We can then write the total energy of a given many-electron system as
where is the complementary error function. As in KS DFT then, minimization is performed over the wavefunction ,
where is the density corresponding to . Eq. (9) yields an effective, long-range-only-interacting hamiltonian to be solved with a chosen wavefunction method. The short-range exchange-correlation functional is then defined as the energy needed to make Eq. (8) exact,
For instance, the correct LSD approximation to is
where is the exchange-correlation energy per electron of the standard uniform electron gas (with Coulomb electron-electron interaction) and is the exchange-correlation energy per electron of a uniform electron gas with interaction Paziani et al. (2006).
where is the spherically and system-averaged pair density (intracule density) obtained by integrating over all variables but ,
with . The gaussian damping appearing in Eq. (12) comes from the derivative of the long-range interaction with respect to , and shows that the exchange-correlation energy is determined by the short-range part of the intracule density. Notice that when Eqs. (12) yields the KS exchange-correlation energy functional from a nonlinear adiabatic connection Savin (1996); Yang (1998).
Iii The extended Overhauser model
The extended Overhuaser model consists in writing an effective Schrödinger-like equation for the intracule density of a given system. The basic idea is the following Gori-Giorgi and Savin (2005); Gori-Giorgi and Savin (2006a, 2007b). We start with the observation that the intracule density couples to any electron-electron interaction operator depending only on the interelectronic distance, , in the same way as the density couples to any local one-body potential operator , i.e.,
We can then follow the Hohenberg and Kohn philosophy but with the roles of and , and of and , interchanged. That is, in analogy with Eq. (2) we can define a system-dependent functional ,
so that the total energy of a given physical system is equal to
Like Kohn and Sham, we can define another functional by setting equal to zero in Eq. (16),
The functional corresponds to the internal kinetic energy of a free (zero external potential) cluster of fermions having the same intracule density of the physical system. The fermions of this cluster interact with an effective interaction which has the same role of the KS potential for the KS system. In practice, this effective interaction must be approximated. Moreover, for electrons the cluster equation become a complicated many-body problem, so that other approximations are needed. As in the original Overhauser model for the uniform electron gas Overhauser (1995); Gori-Giorgi and Perdew (2001), we can approximate the cluster equation with a set of radial geminals ,
whose occupancies must be defined (e.g., in a determinantal-like way as in the original Overhauser model Gori-Giorgi and Perdew (2001)). In practice, trying to solve the whole many-electron Schrödinger equation by means of Eqs. (17)-(19) is a daunting task. The idea is rather Gori-Giorgi and Savin (2005); Gori-Giorgi and Savin (2006a, 2007b) to couple this “average-pair-density-functional theory” with a density functional scheme: Eqs. (19) can be generalized to any along the adiabatic connection of DFT. In Refs. Gori-Giorgi and Savin (2005); Gori-Giorgi and Savin (2007b) we started from the effective interaction which, when inserted in Eqs. (19), gives the intracule density corresponding to the Kohn-Sham system, (that can be obtained from the KS determinant). We then wrote an approximation for along the long-range adiabatic connection of DFT as
The only term that needs to be approximated is then , an effective interaction that should essentially “tell” to the intracule density that, while the electron-electron interaction is turned on (i.e. as increases), the one-electron density does not change. As the information on has been “washed away” in the integration over the center of mass of Eq. (13), this constraint can be imposed only in an approximate way. For two-electron atoms, for which Eq. (19) is exact with one geminal Gori-Giorgi and Savin (2006a), , a simple approximation for is Gori-Giorgi and Savin (2005); Gori-Giorgi and Savin (2007b, a)
where is a screening length associated to the radius of a sphere containing on average one electron Overhauser (1995); Gori-Giorgi and Perdew (2001); Gori-Giorgi et al. (2008). The physical idea behind Eq. (21) is to mimic the constraint of fixed one-electron density by screening the electron-electron interaction over a length associated to the “space” available to each electron (which is determined by the density). Indeed, for the He isoelectronic series Eqs. (12), (19) and (21), combined self-consistently with the Kohn-Sham equations, recover the full CI total energy within 1 mH Gori-Giorgi and Savin (2005); Gori-Giorgi and Savin (2007b, a).
Iv The Overhauser model for the H molecule: how things should be
For a closed-shell physical electronic system (atom, molecule) with particles, the Schrödinger equation describing the internal degrees of freedom of a cluster of fermions having the same intracule density is exactly given by Gori-Giorgi and Savin (2006a, 2007b, 2007a)
As a first study, we calculate and analyze the “exact” Overhauser interaction at full coupling strength (i.e., for electron-electron interaction , corresponding to ) for the H molecule at different values of the internuclear distance , and we compare it with the approximation of Eq. (21). To this purpose, we need extremely accurate intracule densities , which are described in the next Subsec. IV.1.
iv.1 Intracule densities from accurate variational wavefunctions
where and are centers that lie on the internuclear axis, means permutation of and , and is the inversion operator with respect to the center of the molecule. The parameters appearing in Eqs. (23)-(24) are determined variationally by minimizing the energy with the conjugate gradient method (for more details on the wavefunction and the algorithms employed, see Refs. Rychlewski et al. (1994); Cencek et al. (1995); Cencek and Kutzelnigg (1996)). The expansion length in Eq. (23) is used, resulting in energies with the extraordinary accuracy of Hartree.
The intracule densities from these extremely accurate wavefunctions can be easily calculated, since all the needed integrals are analytic. We also calculated the one electron densities , and the intracule densities corresponding to the KS system, which can be obtained by inserting in Eq. (13) the KS wavefunction . In Fig. 1 we show the intracule densities and for the internuclear distances , , and a.u. Although mathematically the wave function of Eqs. (23)-(24) is cuspless, we see that the very elaborate ansatz permits to describe the exact linear behaviour of the intracule density for , up to extremely short distances. Fig. 2 shows the same quantities multiplied by the volume element . This figure better visualizes the transition from dynamical to static correlation. In Fig. 3 we also report the same quantities in the extreme stretched case, , obtained from the simple Heitler-London wavefunction.
iv.2 Accurate Overhauser potentials
From the accurate intracule densities of the previous subsection we can calculate, by inversion, the corresponding “exact” Overhauser interaction ,
The inversion of Eq. (25) is done numerically, by finite differences. In Fig. 4 we report the effective Overhauser interactions that, when inserted in Eq. (22), give the physical and the KS intracule, corresponding, respectively, to and along the long-range adiabatic connection of Sec. II (or to and along the usual linear adiabatic connection in which is simply multiplied by ). We see that for large goes to the same constant for both the KS and the physical system, as it should be Gori-Giorgi and Savin (2007b) (of course if we go to too large we start to observe the wrong harmonic wall due to the gaussian asymptotic decay of our wavefunction). The difference between the effective Overhauser interaction for the physical and the KS system gives of Eq. (20), and is reported in Fig. 5, where also the Coulomb repulsion is shown. From this figure, we see that, when the system is still dominated by dynamical correlation, as in the case and , is essentially a screened Coulomb interaction. That is, for short-range it behaves as , and then for large goes to zero much faster than . In such cases, the approximation of Eq. (21), which at reads
can work reasonably well, with a screening length . However, as grows and the system starts to be dominated by static correlation, we see that the approximation of Eq. (26) cannot work: the “exact” still decays much faster than for large , but at short range is more repulsive than the Coulomb interaction! I.e., we need an “overscreened” interaction. This is completely evident in the extreme stretched case of Fig. 6, again obtained from the simple Heitler-London wavefunction.
V Generalized optimized effective potential method for multideterminant DFT
In recent years, the focus of a large part of the scientific community working on improving the approximations for has shifted from seeking explicit functionals of the density (like the generalized gradient approximations), to implicit functionals, typically using the exact exchange , which is only explicitly known in terms of the Kohn-Sham orbitals . The corresponding Kohn-Sham potential must then be computed with the optimized effective potential (OEP) method (for a recent review, see Kümmel and Kronik (2008)). The OEP scheme can be generalized to the multideterminant range-separated DFT by first noticing that we can divide into exchange and correlation in two different ways Toulouse et al. (2005a): we can define the exchange energy with respect to the KS determinant ,
and then define the usual correlation energy functional as the energy missed by the KS wavefunction,
but we can also define a multideterminantal (md) exchange functional Toulouse et al. (2005a) by using the wavefunction ,
and then a corresponding correlation energy that recovers the energy missed by (which is smaller than the energy missed by the KS determinant ),
Then, with this latter definition of the correlation energy, the generalized OEP-like scheme for multideterminant DFT becomes Toulouse et al. (2005a)
where is obtained by solving the Schrödinger equation corresponding to the hamiltonian
Notice that this multideterminant OEP scheme is different from the one recently proposed in Ref. Weimer et al. (2008). In Eq. (31) the weak long-range interaction automatically selects the configuration space needed to yield an accurate solution for the hamiltonian of Eq. (32), while in Ref. Weimer et al. (2008) the configuration space is chosen essentially by hand, using physical and chemical intuition.
In the next Sec. VI we use the Overhauser model to approximate , and we apply our combined formalism to the case of the H molecule.
Vi Multideterminant DFT combined with the Overhauser model
From the adiabatic connection formalism we can easily write an exact formula for ,
which shows that is determined by the change in the short-range part of the intracule density when the electron-electron interaction increases from to the full Coulomb repulsion . By adding and subtracting , Eq. (33) can also be written as
where is the correlation energy of Eq. (28), defined with respect to the KS determinant.
We computed within the Overhauser model, Eq. (19) with one geminal , using the simple screened potential of Eqs. (20)-(21) with the screening length . For each internuclear distance , the intracules have been calculated for 33 values of between and . By numerical integration we then computed
and we fitted the values of with the derivative of the function
We then implemented the generalized OEP scheme of Eq. (31) by first minimizing the effective potential at the “generalized-exchange”-only level, and by adding only as a final correction. Since is very small, we do not expect substantial changes by implementing a full self-consistent scheme. Our procedure can be summarized with the equation
where is calculated with the final density resulting from the minimization in the first term on the right-hand-side of Eq. (37).
To carry out the minimization with respect to the potential in Eq. (37) we proceeded as follows. We parametrized the potential with a simple two-parameter form, by adding to the physical external potential a gaussian centered on each atom, . The minimization of the expectation with respect to the two parameters and is done by calculating at each step full-CI wavefunctions for the hamiltonian with electron-electron interaction and external potential . All calculations were done at the cc-V5Z basis-set level. We also produced with MOLPRO Amos et al. (2002) full CI reference results for the physical hamiltonian, for comparison. Our simple parametrization of the potential , containing only two parameters, is enough to yield at the HF energy within 0.5 mH, which is the accuracy we sought in this study. This way, we avoid all the well-known problems of the OEP method in finite basis set Staroverov et al. (2006) at the price of obtaining only an upper bound for our minimization problem (yet, with the reasonable accuracy of 0.5 mH).
In Fig. 7 we report the results for for three different values of the internuclear distance . The dots () are the “exact” values of , i.e., the full-CI total energies obtained with MOLPRO minus the energies corresponding to the first term on the right-hand-side of Eq. (37), . The solid line is from the Overhauser model, and the dashed line is the LDA result, obtained from the parametrization of Ref. Paziani et al. (2006), in which for the uniform electron gas has been calculated with Quantum Monte Carlo methods. We see from this figure that when the system is still dominated by dynamical correlation, as in the and the cases, the Overhauser model yields, even at (i.e. for pure KS DFT), correlation energies with errors of mH (while LDA is off by mH), which reduce to 1 mH at (where the LDA error is still mH). We focuse here on the value since it is the one commonly used in practical applications Ángyán et al. (2005); Toulouse et al. (2005a); Goll et al. (2005, 2006); Gerber et al. (2007); Fromager et al. (2007); Fromager and Jensen (2008). When the system starts to be dominated by static correlation, as in the case, the Overhauser model with the simple screened potential of Eq. (21) gives, at , errors very close to those of LDA ( mH), which are still of the order of mH at . As the molecule approaches the dissociation limit, , the exact tends to the limiting behavior in which , and for any . This is due to the fact that, as , the long-range only wavefunction , even at very small (i.e., with an infinitesimal interaction), becomes essentially exact and equal to the Heitler-London wavefunction, so that the functional should be just equal to zero. In this limit, the Overhauser model is wrong for small (because, as explained in Sec. IV, it misses the “overscreening” at short range), but, for , yields that go to zero much faster than LDA, as it can be already grasped from the third panel of Fig. 7. It is thus still more suitable than LDA to be combined with the range-separated multideterminant DFT, but it definitely needs some improvement.
Notice that the Overhauser model would yield much more accurate results if we were able to compute by using in Eq. (20) instead of the interaction which yields the intracule associated to the wavefunction . This way, we would use the information available in to the maximum extent, and we would not have the problems associated to the “overscreening” discussed in Sec. IV. This possibility will be investigated in future work.
Vii Conclusions and perspectives
We have presented a preliminary study of the combination of range-separated multideterminant DFT with the Overhauser model, with an application to the paradigmatic case of the H molecule. We have first analyzed, by means of very accurate variational wavefunctions, the failure of the Overhauser model in describing static correlation and we have then used it to produce an adapted short-range correlation functional for range-separated multideterminant DFT. The results are very good for internuclear distances close to equilibrium, and are still encouraging as the molecule is stretched. Indeed, in the dissociation limit the exact short-range correlation functional should go to zero for any , and the Overhauser model yields short-range correlation energies that go to zero faster than LDA as increases.
Future work will address the study of better approximations for the unknown Overhauser electron-electron interaction, and the development of a more efficient scheme to combine it with range-separated multideterminant DFT.
We thank W. Cencek for providing us with the accurate geminal wave functions of the H molecule. This work was supported by the ANR (National French Research Agency) under Grant n. ANR-07-BLAN-0271.
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