Uniform Electron Gases.
II. The Generalized Local Density Approximation in One Dimension
We introduce a generalization (gLDA) of the traditional Local Density Approximation (LDA) within density functional theory. The gLDA uses both the one-electron Seitz radius and a two-electron hole curvature parameter at each point in space. The gLDA reduces to the LDA when applied to the infinite homogeneous electron gas but, unlike the LDA, is is also exact for finite uniform electron gases on spheres. We present an explicit gLDA functional for the correlation energy of electrons that are confined to a one-dimensional space and compare its accuracy with LDA, second- and third-order Møller-Plesset perturbation energies and exact calculations for a variety of inhomogeneous systems.
pacs:71.10.Ca, 31.15.V-, 02.70.Ss
I Local Density Approximation
The local density approximation (LDA), unlike most of the “sophisticated” density functional approximations in widespread use today, is truly a first-principles quantum mechanical method.Parr and Yang (1989) It is entirely non-empirical, depending instead on the properties of one of the great paradigms of modern physics: the infinite homogeneous electron gas (HEG).Fermi (1926); Thomas (1927) Application of the LDA is straightforward, at least in principle. Although the electronic charge density in any real system is non-uniform, the LDA proceeds by assuming that the charge in an infinitesimal volume element around the point behaves like a locally homogeneous gas of density , and adds all of the resulting contributions together. This implicitly assumes that the infinitesimal contributions are independent (which is undoubtedly not the case) but then requires only that the properties of the HEG be known for all values of .
The density of a HEG is commonly given by (the number of electrons per unit volume) or the Seitz radius and these equivalent parameters are related by
where is the dimensionality of the space in which the electrons move. In terms of these, the LDA correlation functional is
where the correlation kernel is the reduced (i.e. per electron) correlation energy of the HEG with Seitz radius .
In high-density HEGs (i.e. ), the kinetic energy dominates the Hamiltonian and the Coulomb repulsion between the electrons can be treated via perturbation theory. This has facilitated investigations of in 3DWigner (1934); Macke (1950); Bohm and Pines (1953); Pines (1953); Gell-Mann and Brueckner (1957); DuBois (1959); Carr, Jr. and Maradudin (1964); Misawa (1965); Onsager, Mittag, and Stephen (1966); Isihara and Kojima (1975); Kojima and Isihara (1976); Wang and Perdew (1991); Hoffman (1992); Endo et al. (1999); Ziesche and Cioslowski (2005); Loos and Gill (2011a) and 2DZia (1973); Glasser (1977); Rajagopal and Kimball (1977); Isihara and Toyoda (1977, 1978); Isihara and Ioriatti (1980); Glasser (1984); Seidl (2004); Chesi and Giuliani (2007); Loos and Gill (2011b) but, because the Coulomb operator is so strong in 1D that two electrons cannot touch, the 1D gas has received less attention.Fogler (2005); Astrakharchik and Girardeau (2011); Loos (2013)
In low-density HEGs (i.e. ), the potential energy dominates, the electrons localize into a Wigner crystal and strong-coupling methods can be used to find asymptotic expansions of . Here, too, the 3D,Coldwell-Horsfall and Maradudin (1960); Carr, Jr. (1961); Carr, Jr., Coldwell-Horsfall, and Fein (1961) 2DMeissner, Namaizawa, and Voss (1976); Bonsall and Maradudin (1977) and 1DFogler (2005) HEGs have all been studied.
For intermediate densities, the best estimates of come from Quantum Monte Carlo (QMC) calculations, as pioneered by Ceperley and refined by several other groups.Ceperley (1978); Ceperley and Alder (1980); Tanatar and Ceperley (1989); Kwon, Ceperley, and Martin (1993); Ortiz and Ballone (1994); Rapisarda and Senatore (1996); Kwon, Ceperley, and Martin (1998); Ortiz, Harris, and Ballone (1999); Varsano, Moroni, and Senatore (2001); Foulkes et al. (2001); Attaccalite et al. (2002); Zong, Lin, and Ceperley (2002); Mitas (2006); Drummond and Needs (2009); Lüchow, Petz, and Schwarz (2010); Shepherd et al. (2012) By combining these with the high- and low-density results, various groupsVosko, Wilk, and Nusair (1980); Perdew and Zunger (1981); Perdew and Wang (1992a); Sun, Perdew, and Seidl (2010) have constructed interpolating functions that allow to be estimated rapidly for any value of .
Unfortunately, this approach is flawed, for the correlation energy of a uniform electron gas depends on more than just its value.Gill and Loos (2012) We have therefore argued that should be generalized to , where the parameter measures the two-electron density. Although not mathematically mandated,Pérez-Jiménez et al. (2001) we prefer that , like , be a local quantity. In Section II, we propose a definition for inspired by a number of previous researchers.Colle and Salvetti (1975); Stoll, Golka, and Preuss (1980); Becke (1983); Luken and Culberson (1984); Dobson (1991)
To learn more about the two-parameter kernel, we have turned to the finite uniform electron gases (UEGs) formed when electrons are confined to a -sphere.Loos and Gill (2009a, b, 2010a, 2010b, 2010c, 2011c, 2012a, 2013) In Section III, we report accurate values of and for electrons on a 1-sphere, systems that we call “-ringium”. In Section IV, we devise three functionals to approximate these results and in Section V, we test two of these on small 1D systems. Atomic units are used throughout.
Ii Hole curvature
Suppose that an electron lies at a point . The probability that a second electron lies at is givenCoulson and Neilson (1961); Coleman (1967); Colle and Salvetti (1975); Stoll, Golka, and Preuss (1980); Perdew and Wang (1992b); Cioslowski and Liu (1998); Lee and Gill (1999); Gill et al. (2000); Gill, O’Neill, and Besley (2003); Gill et al. (2006); Gill (2011); Proud, Walker, and Pearson (2013) by the conditional intracule
where is the exchange-correlation holeParr and Yang (1989) and
is the spinless second-order density matrix.Davidson (1976) For fixed , we have the normalization
Because the Laplacian measures the tightness of the hole around the electron at and has dimensions of , we can use the dimensionless hole curvature
to measure the proximity of other electrons to one at . (We will fix the coefficient in the next Section.) It is difficult to find this Laplacian for the exact wave function but, at the Hartree-Fock (HF) level, it involves simple sums over the occupied orbitals, viz.
and we will therefore employ HF curvatures henceforth.111In response to a referee, we have calculated the exact and HF curvatures in 2-ringium for several values of the ring radius . The HF curvature is for all . In contrast, the exact curvature diminishes from at , to at , to at , and finally to at . The interesting connection between the curvature and the kinetic energy density Becke (1983) is worth noting.
Iii Calculations on -Ringium
iii.1 Density and curvature
and, because of the symmetry of the system, the density and Seitz radius
If we choose so that for the 1D HEG (i.e. -ringium), we obtain
In general, requiring that in the -dimensional HEG leads (via Fermi integration) to
and the particular values and .
iii.2 Correlation energy
The Hamiltonian for electrons on a ring is
where is the distance (across the ring) between electrons and . As noted previously,Loos and Gill (2013) the energy is independent of the spin-state and so we assume that all electrons are spin-up. The exact wave function can then be written as , where the correlation factor
|Degree 0||Degree 1||Degree 2||Degree 3|
Judicious integration by parts allows us to partition the total energy
into the HF energyLoos and Gill (2013)
and the correlation energy
can be minimized either by QMC methodsLüchow, Petz, and Schwarz (2010) or via the secular equation
where the overlap, kinetic and Coulomb matrix elements
can be found analytically in Fourier space (Appendix A). We have used the CASINO QMC packageNeeds et al. (2010) and, where possible, the Knowles–Handy Full CI program to confirm results.Knowles and Handy (1984, 1989)
Table 2 shows the resulting near-exact correlation energies for ground-state -ringium. (Where these energies differ from those in Table VI of Ref. Loos and Gill, 2013, the new values are superior.) The fact that the values in a given column are not equal demonstrates that the correlation energy of a UEG is not determined by its value alone.Gill and Loos (2012) Moreover, the variations in for a given are large: the values, for example, are only about half of the values, implying that the correlation energy of a few-electron system is grossly overestimated by the LDA functional which is based on the HEG.
Iv Generalized Local Density Approximation
In the LDA, the correlation contribution is estimated from alone, according to Eq. (2). However, the fact that UEGs with the same , but different , have different energies compels us to devise a Generalized Local Density Approximation (GLDA) wherein we write
where the correlation kernel is the reduced correlation energy of a UEG with Seitz radius and curvature . For present purposes, we will use and values from the HF, rather than the exact, wave function.
One might think that the kernel could be constructed by fitting the results in Table 2 but these data allow us to construct only for . To construct the rest of the kernel will require accurate correlation energies for uniform gases with high curvatures () but, although these arise in excited states of -ringium, this raises some fundamental questions which lie outside the scope of the present manuscript and will be discussed elsewhere.
iv.1 High densities
Rayleigh-Schrödinger perturbation theory for -ringium yields the high-density expansion
The leading coefficientLoos and Gill (2013) is
but, if we fit a truncated version of this series, while ensuring that vanishes for one electron, we obtain the approximation
which can be rewritten in terms of the curvature, using Eq. (13) to obtain
The accuracy of this approximation is shown in columns 2 and 3 of Table 3.
iv.2 Low densities
Strong-coupling perturbation theory for -ringium yields the low-density expansion
The leading coefficient is the difference between the Wigner crystal Coulomb coefficient
and the HF Coulomb coefficient
It follows that
but, if we truncate this series after the term and modify it to ensure that vanishes for one electron, we obtain the approximation
which can be rewritten in terms of the curvature, using Eq. (13) to obtain
The accuracy of this approximation is shown in columns 4 and 5 of Table 3.
iv.3 Intermediate densities
How can we model for fixed ? Ideally, we would like a function that reproduces the behaviors of Eqs (24) and (28) and interpolates accurately between these limits. However, for practical reasons, we will content ourselves with a function that approaches for small , behaves like for large , and changes monotonically between these.
possesses all of the desired features and we therefore adopt the approximate kernel
reproduces the Table 2 data to within a relative error of 1% and absolute error of 0.20 .
iv.4 The LDA1, GLDA1 and gLDA1 functionals
We can now consider three approximate kernels for correlation in 1D systems. The first is the LDA1 kernel, which is defined by
where , and . This underpins the traditional LDA and, by construction, it is exact (within fitting errors) for the 1D HEG or, equivalently, for -ringium. It is independent of the hole curvature .
The second is the GLDA1 kernel, which is defined by
where , and are defined in Eqs (27), (33) and (38). Unfortunately, because of a lack of information about high-curvature UEGs, these three equations are not defined for and thus, at this time, the GLDA1 is defined only for systems where at all points. Completing the definition of the GLDA1 is an important topic for future work.
The third is the gLDA1 kernel, a partially corrected LDA, which is defined by
When applied to UEGs with , the gLDA1 and LDA1 kernels are, of course, identical. However, when applied to gases with , they behave differently and, by construction, the gLDA1 kernel is exact (within fitting errors) for any -ringium.
The gLDA1 kernel defaults back to the LDA1 kernel at points where but we cannot predict a priori whether this will cause it to under-estimate or to over-estimate the GLDA. If the monotonic increase in the magnitude of the kernel between and continues beyond , then the gLDA1 kernel (which assumes that the kernel is constant beyond ) will underestimate the GLDA1 kernel and consequently underestimate the magnitude of the correlation energies in systems with high-curvature regions.
Until the true kernel for is known, we cannot draw any firm conclusions about the accuracy of GLDA1. However, it is reasonable to conjecture that even the imperfect gLDA1 may be superior to LDA1 for density functional theory (DFT) calculations on inhomogeneous 1D systems and we now explore this through some preliminary validation studies.
Having defined the gLDA1 functional, we turn now to its validation. The functional is exact by construction for any -ringium, so we require systems with non-uniform densities. There is no standard set of 1D models with accurately known correlation energies, so it was necessary to devise our own and we chose the ground states of electrons in a 1D box of length (a family that we call the -boxiums) and of electrons in a 1D harmonic well with force constant (a family that we call the -hookiums). Whereas the HOMO–LUMO gap in -boxium increases roughly linearly with , that in -hookium slowly decreases. We therefore regard them as “large-gap” and “small-gap” systems, respectively.
Given that the fitting errors (Table 3) in the gLDA1 functional can be of the order of 0.1 , we aimed to obtain the energies of the -boxium and -hookium to within 0.1 of their complete basis set (CBS) limits. This is easily achieved for the HF, LDA1 and gLDA1 energies, because they converge exponentiallyKutzelnigg (1994); McKemmish and Gill (2012); Kutzelnigg (2013) with the size of the one-electron basis, but it is less straightforward for traditional post-HF energies.
We analysed the convergence behavior (see Appendix B) of Møller-Plesset perturbation (MP2 and MP3) and full configuration interaction (FCI) energies in 2-ringium, 2-boxium and 2-hookium and our results are summarised in Table 4. From these, we devised appropriate extrapolation formulae and applied these to the energies obtained with our largest basis sets. We also used QMC calculationsNeeds et al. (2010) to assess the accuracy of our extrapolated FCI energies.
Tables 5 and 6 show the energies obtained for 5-boxium and 5-hookium, respectively, as the basis set size increases from to . The three components of the third-order energy are separated because of their different convergence behaviors. Table 7 summarizes our best estimates of the HOMO–LUMO gaps, together with the HF, LDA1, gLDA1, MP2, MP3 and FCI energies, for -boxium and -hookium with = 2, 3, 4 or 5.
The 2-boxium system (albeit with length ) was studied in a basis of delta functions by Salter et al. Salter, Trucks, and Cyphert (2001) and, using 804609 basis functions, they obtained energies within roughly 10 of the exact values. The present work is the first study of -boxium with .
The orbitals of 1-boxium are
and the first of these form a convenient orthonormal basis for expanding the HF orbitals in -boxium. The antisymmetrized two-electron integrals can be found in terms of the Sine and Cosine Integral functionsOlver et al. (2010) and we have used these to perform SCF calculations with up to basis functions. Our convergence criterion was .
We first discuss 2-boxium. Choosing yields the HF orbitals
and Fig. 1 reveals that the density has maxima at , indicating that an electron is likely to be found in these regions. LDA1 interprets these maxima as the most strongly correlated regions in the well and, through Eqs (2) and (39), predicts the correlation energy
In contrast, because the hole curvature is strongly peaked at the center and edges of the box and is small near the density maxima, gLDA1 identifies the center of the box as the most correlated region and Eqs (23) and (41) predict the much smaller correlation energy
LDA1 and gLDA1 offer very different qualitative and quantitative descriptions of 2-boxium, but both perturbation theory ( and ) and near-exact calculations () support the gLDA1 picture.
We have also performed HF, LDA1, gLDA1, MP2, MP3 and FCI calculations on 3-, 4- and 5-boxium and the density and curvature for 5-boxium are shown on the right of Fig. 1. Both functions oscillate much more rapidly but with much smaller amplitude than in 2-boxium, and it is easy to foresee that, as the number of electrons becomes large, both the density and the curvature will become increasingly uniform.
The convergence of the 5-boxium energies is shown in Table 5 and confirms the theoretical predictions of Table 4. The LDA1 energies, which depend only on the density , converge rapidly, changing by less than 1 beyond . The HF and gLDA energies, which depend on the orbitals (rather than the density) converge more slowly, achieving 1 convergence around . Because the occupied orbitals converge more rapidly than the virtual ones,Deng and Gill (2011) the component of MP3 converges almost as fast as HF, the component (which is negative) converges more slowly, and the component (which is positive) even more slowly.222The symbols “O” and “V” refer to the number of occupied and virtual orbitals, respectively. The component, for example, involves four sums over occupied orbitals and two over virtual orbitals. Because of the resulting differential cancellation,Noga and Kutzelnigg (1994); Köhn and Tew (2010) the total 3rd-order contribution initially becomes more negative, reaches a minimum at and rises thereafter. The MP2 energy is the most slowly converging, and changes by 60 between and . It is interesting to note the almost perfectly linear growth of the third-order energies. Because the -boxiums are large-gap systems, MP2 and MP3 work well, recovering more than 92% and 99% of the correlation energy in 5-boxium.
Our best estimates of the CBS limit HF and correlation energies are summarized in the left half of Table 7. Because LDA1 operates without the benefit of curvature information, it gravely overestimates the correlation energy, by between a factor of five (for 2-boxium) and a factor of just under two (for 5-boxium). In contrast, gLDA1 is within 12% of the true correlation energy for all -boxiums studied.
Electrons in 3D harmonic wells have been studied by numerous authorsKestner and Sinanoğlu (1962); Santos (1968); White and Byers Brown (1970); Benson and Byers Brown (1970); Kais, Herschbach, and Levine (1989); Taut (1993); Ivanov, Burke, and Levy (1999); Cioslowski and Pernal (2000); Henderson, Runge, and Bartlett (2001); O’Neill and Gill (2003); Katriel, Roy, and Springborg (2005); Gill and O’Neill (2005); Ragot (2008); Loos and Gill (2009c); Cioslowski (2013) but this is the first investigation of electrons in a 1D harmonic well. The orbitals of 1-hookium are
and the first of these form a convenient orthonormal basis for expanding the HF orbitals in -hookium. The antisymmetrized two-electron integrals can be found in closed form (e.g. see Appendix B) and we have used these to perform SCF calculations with up to basis functions. Our convergence criterion was .
We first discuss 2-hookium. Choosing yields the HF orbitals
and Fig. 2 reveals that the density and curvature are softened versions of those in 2-boxium. As before, LDA1 interprets the density maxima as regions of strong correlation, predicting
whereas gLDA1 finds that almost all of the correlation comes from a narrow region near the middle of the well and predicts
As for 2-boxium, LDA1 and gLDA1 offer entirely different pictures of electron correlation but both perturbation theory ( and ) and near-exact calculations () agree that gLDA1 is closer to the truth.
We have also performed HF, LDA1, gLDA1, MP2, MP3 and FCI calculations on 3-, 4- and 5-hookium and the density and curvature for 5-hookium are shown on the right of Fig. 2. As before, both functions oscillate more rapidly but with smaller amplitude than in 2-hookium and it is clear that, as the number of electrons becomes large, both functions will become increasingly uniform.Loos and Gill (2012b)
|-boxium ()||-hookium ()|
The convergence of the 5-hookium energies is shown in Table 6. As in 5-boxium, the LDA1 energies converge most rapidly, followed by the HF and gLDA1 energies, then the , and components of the third-order energy, and finally the MP2 energy. However, each of these energies converges significantly more slowly than its 5-boxium analog. All of these observations are consistent with the theoretical predictions of Table 4. Because the -hookiums are smaller-gap systems, MP2 and MP3 are less successful than for -boxium, recovering roughly 85% and 96% of the correlation energy in 5-hookium.
Our best estimates of the CBS limit HF and correlation energies are summarized in the right half of Table 7. As before, whereas LDA1 seriously overestimates the correlation energies, gLDA1 is within 15% of the true correlation energy in all cases. It is interesting to note that (-hookium) (-boxium) in all cases but that, whereas gLDA1 correctly predicts this trend, LDA1 reverses it.
Vi Concluding Remarks
The traditional Local Density Approximation (LDA) is exact by construction for an infinite uniform electron gas with Seitz radius . However, it significantly overestimates the magnitudes of correlation energies in finite gases, such as those created when electrons are placed on the surface of a -dimensional sphere. This overestimation, which becomes even more pronounced in non-uniform gases, led us to seek generalizations of the LDA which are exact for both infinite and finite gases and, in the present work, we have proposed that the local hole curvature provides the necessary information to achieve this goal. For present purposes, we have extracted from the HF wave function: this requires only the occupied HF orbitals.
By fitting accurately calculated correlation energies for systems of electrons on a ring, we have constructed the Generalized Local Density Approximation for one-dimensional systems and this has yielded a correlation kernel and a corresponding functional which we call GLDA1. To this point, we have considered only gases in which and, consequently, the GLDA1 functional is not yet defined for gases with higher curvature. However, if we assume that the the correlation kernel becomes flat, i.e. that when , we obtain an approximation to GLDA1 which we call gLDA1.
We have applied the traditional LDA1 functional and the curvature-corrected gLDA1 functional to electrons trapped in 1D boxes or in 1D harmonic wells and, by comparing the predicted correlation energies with those obtained from MP2, MP3 and Full CI calculations, we have discovered that gLDA1 is much more accurate than LDA1 in all cases.
We have also observed that gLDA1 tends to underestimate the magnitudes of correlation energies. This suggests that the true GLDA1 kernel continues to rise, i.e. that but systematic examination of high-curvature () gases is required to test this. Such exploration is an important topic for future research and will allow the GLDA1 functional to be completely defined and tested.
Although we have presented relatively few calculations here, and much more investigation is warranted, these preliminary results suggest that “curvature-corrected density functional theory (CC-DFT)” may offer an efficient pathway to improvements over existing functionals.
Acknowledgements.P.F.L. and P.M.W.G. thank the NCI National Facility for generous grants of supercomputer time. P.M.W.G. thanks the Australian Research Council (Grant Nos. DP0984806, DP1094170 and DP120104740) for funding, P.F.L. thanks the Australian Research Council for a Discovery Early Career Researcher Award (Grant No. DE130101441) and C.J.B. is grateful for an Australian Postgraduate Award. P.F.L. and P.M.W.G. thank Neil Drummond and David Tew for helpful discussions and P.M.W.G. thanks the University of Bristol for sabbatical hospitality during the construction of this manuscript.
Appendix A: Calculation of matrix elements
The matrix elements in Eq. (22) are expressed as expectation values of operators over the HF wave function. Therefore, because and its reduced density matrices, e.g.
have finite Fourier expansions, integrals of their products with Fourier expansions of operators reduce to finite sums.
The Fourier expansions of bounded operators on a unit ring are straightforward, e.g.
The expansions of unbounded operators, e.g.
are delicate (they converge only in the Cesàro meanOlver et al. (2010)) but this is sufficient for our purposes because we require only a few of the low-order Fourier coefficients. The expansions of “cyclic” operators (e.g. ) are not simple products and must be derived separately.
Thus, for example, to find the integral in 3-ringium, the Fourier expansion
is combined with Eq. (53) to yield
Appendix B: Extrapolation of perturbation energies
It is common these days to estimate the CBS limit of post-HF correlation energies by extrapolation.Helgaker et al. (1997) Pioneering work by Schwartz,Schwartz (1962) HillHill (1985) and Kutzelnigg and MorganKutzelnigg and Morgan III (1992) showed that, for atoms in 3D, the second-order energy contributions from basis functions with angular momentum converge asymptotically as .
While generating the data in Section V, we found that the MP2 and MP3 energies converge so slowly (Tables 5 and 6) that the CBS limit is not reached (within our 0.1 target accuracy), even with our largest () basis set. This is particularly noticeable for -hookium. We therefore needed to develop and apply appropriate extrapolation procedures.
To this end, we analyzed the convergence of the second-order energy
obtained from the non-interacting orbitals and orbital energies in 2-boxium and 2-hookium. In -hookium, the double-bar integral is
if is odd but it vanishes if is even. The orbital energies are given by . By substituting these expressions into (58) and making use of Stirling’s approximation,Olver et al. (2010) one can show that the error introduced by truncating the basis after functions is
The closed-form expression for the integral in -boxium is cumbersome but a similar analysis reveals that the analogous truncation error is . The truncation errors in the third-order energy can be found in the same way and all of our results are summarized in Table 4.
The MP2, MP3 and FCI energies obtained with our largest basis sets conform to these analytical predictions and allowed us to extrapolate reliably to the CBS energies given in Table 7. The good agreement between our extrapolated FCI energies and QMC energies further increases our confidence in these results.
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