Understanding electron correlation energy through density functional theory

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1 1 Understanding electron correlation energy through density functional theory by Teepanis Chachiyo, 1,2,* and Hathaithip Chachiyo 2 1 Department of Physics, Faculty of Science, Naresuan University, Phitsanulok Thailand 2 Thailand Center of Excellence in Physics, Commission on Higher Education, 328 Si Ayutthaya Road, Bangkok 10400, Thailand * Correspondence to: <teepanisc@nu.ac.th> Summary What does correlation energy really mean? Density functional theory (DFT) enumerates the total energy of electrons into five contributions: 1) the kinetic energy as electrons move in space, 2) the potential energy that binds electrons to nuclei, 3) the Coulomb repulsion energy as electrons repel each other, 4) the exchange energy due to the Pauli exclusion principle, and 5) the correlation energy. Although DFT has become the most widely used theory in describing behaviors of molecules and bulk material, an intuitive meaning of the fifth contribution has remained elusive for over 90 years in the history of quantum mechanics. Here, we hypothesize that the correlation energy is the energy that tries to keep electrons in order, evenly distributed in a uniform fashion. As such, the energy assumes its maximum value when a gradient of density is zero. As the gradient increases, the energy is diminished by a gradient suppressing factor, designed to attenuate the energy from its maximum value similar to the shape of a bell curve. Based on this hypothesis we constructed a very simple mathematical formula that predicted the correlation energy of atoms and molecules. Combined with our proposed exchange energy functional, we calculated the correlation energies, the total energies, and the ionization energies of test atoms and molecules; and despite the unique simplicities, the functionals accuracies are in the top tier performance, competitive to the B3LYP, BLYP, PBE, TPSS, and M11. Therefore, we conclude that, as guided by the simplicities and supported by the accuracies, the correlation energy is the energy that tries to keep electrons in order, for it favors the uniformity of the electrons distribution.

2 2 We know how to compute it, but nobody knows what it really means. The term correlation energy was introduced by Wigner 1 in the 1930s who pointed out its significance in the field of solid state physics. In quantum mechanics, the method of computing the correlation energy is rooted on a perturbation theory 2. For example, when there are many electrons in a molecule, each electron is represented by a molecular orbital, a function describing how the electron is distributed in space. A perturbation theory translates the complex interplay between the electrons into a complex mathematical expression involving the molecular orbitals. The higher the order of the perturbation, the more accurate the energy becomes. In this view, the correlation energy is the 2 nd order onward with exceedingly more complicated expression from one order to the next. Indeed, there are many methods which can calculate the energy very accurately such as the Coupled-Cluster theory 3, Configuration Interaction 2, and Moller-Plesset perturbation theory 4. Through the complexity of theoretical formulations, one may use the molecular orbitals to compute the correlation energy very accurately; but one still does not know what it means. If we want to understand the correlation energy from the perspective of the density functional theory 5, we are forced to sort out its meaning using only the electron density. That is, the meaning of the correlation energy must intimately tie to how the electrons arrange themselves in molecules or bulk material. Therefore, we hypothesize that the correlation energy is the energy that tries to keep electrons in order, evenly distributed in a uniform fashion. The hypothesis leads to a very simple expression as follows. E c = ρε c (1 + t 2 ) h ε c d 3 r Eq. (1) Here, ρ is the total electron density; and ε c is the correlation energy when the density is perfectly uniform, which we use the Chachiyo s formula 6-8 ε c = a ln(1 + b r s + b r s 2). The term (1 + t 2 ) h ε c is the gradient suppressing factor, designed to attenuate the energy from its maximum value as a gradient parameter t = ( π 6 3 )1 a 0 4 ρ increases. In addition, the ρ 7 6

3 constant h = 1 (8.470 2 10 3 ) 16 ( 3 3 π )1 = 0.06672632 Hartree is related to the behavior of the correlation energy when electron density varies very slowly.

3 3 constant h = 1 ( ) 16 ( 3 3 π )1 = Hartree is related to the behavior of the correlation energy when electron density varies very slowly. a Maximum Suppressing factor gradient b Calculated Calculated Correlation Energies (mhartree) H He Li Be B C N O F Ne H2 LiH BeH CH CH4 NH NH3 OH H2O FH CO N2 O2 CO2 using correlation functional This Work Experiment (Ref.16) Average Error This Work 5.3 mh B3LYP 5.3 mh BLYP 18.7 mh PBE 46.4 mh TPSS 34.5 mh M mh This Work Experiment Figure 1 a Diagram showing how the strength of the correlation energy is suppressed as the gradient parameter t increases. b Correlation energies predicted by the functional in this work. Basis set QZP was used throughout. The derivation of Eq. (1) is surprisingly simple. As shown in Fig. 1a) if we conjecture that the correlation energy favors the uniformity of the electron density, then it must assume a maximum value when the gradient parameter t = 0. As t increases, the energy is diminished which we account for it by using a gradient suppressing factor S(t) in a form similar to the shape of a bell curve (Gaussian decay): S(t) = e b ln(1+t2 ) ; Eq. (2)

4 4 where the parameter b controls how rapidly the energy is diminished in response to the increasing gradient. It can be determined by considering the limit t 0 which is the case where the density varies very slowly. In this limit, Ma and Brueckner used a perturbation theory to derive an expression for the correlation energy 9 : E c [ρε c ρ 2 ] ρ 4 3 d3 r Rydberg as ρ Eq. (3) Comparing Eq. (2) and Eq. (3) in this limit t 0, as elaborated in the supplementary information, we have b = h ε c. Hence h ln(1+t S(t) = eε 2 ) c = e ln[(1+t2 ) h ε c ] = (1 + t 2 ) h ε c as indicated in the Eq. (1).

5 5 Table I Total energies (Hartree) Exact (Ref.16) This Work H LiH BeH CH CH NH NH OH H2O FH CO N O CO This Work B3LYP Average 5.0 mh 6.4 mh BLYP 20.3 mh PBE 55.4 mh TPSS 40.1 mh M mh Table II Ionization energies (ev) Exp. (Ref.17) This Work H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar This Work 0.14 ev B3LYP 0.15 ev Avg. BLYP 0.18 ev PBE 0.15 ev TPSS 0.13 ev M ev Fig. 1b), Table I, and Table II illustrate the accuracy of the correlation functional in Eq. (1) via the calculated correlation energies, the total energies, and the ionization energies of atoms and molecules. The performance is comparable to that of B3LYP 10, BLYP 11,12, PBE 13 TPSS 14, and M Considering that the five functionals are among the most cited works of all time 16, the results validate the accuracy of the correlation energy functional in Eq. (1). However, to produce the results as shown one also needs an exchange energy contribution. Even though the behavior of the exchange energy is known for the most part, there is still a few missing pieces one of which we would like to contribute in this work. It is well known how the exchange energy behaves in two opposing limits: 1) when the electron density is perfectly uniform or varies very slowly 19, and 2) when the electron density decays very rapidly 20. Here, we propose an interpolation function which merges the two limits into a

6 H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr %Error 6 complete picture. After a few mathematical arguments as described in the supplementary information, the interpolation leads to a simple and accurate exchange functional as follows. E x = ρε x 3x 2 +π 2 ln(x+1) (3x+π 2 ) ln(x+1) d3 r ; (4) Here, ε x is the Dirac exchange energy 21 for uniform electron gas; and the parameter x = ρ 2 3 ρ (π 3 )1 is used to represent the gradient of the electron density. Tests on hydrogen through krypton atoms are summarized in Fig. 2. As shown, the performance of the exchange functional is highly competitive to the well-established ones such as Becke-88 11, PBE 13, and the meta-gga MVS Atomic Exchange Densities Avg. %Error This Work Becke MVS PBE Spin Multiplicity M=(2S+1) This Work Becke88 MVS PBE Figure 2 Comparisons between this work and the three well-established exchange functionals. The DFT exchange energies were computed at the Hartree-Fock densities using the QZP basis set.

7 7 Owing to the accuracy of the energy functional in Eq. (1), we further discuss the implications of the hypothesis on the meaning of the correlation energy, both qualitatively and quantitatively. Qualitatively, it is well known that the correlation energy plays a pivotal role in chemical bonding. This can easily be explained if one subscribes to the idea that the correlation energy favors uniform electron distribution. In the bonding region, the electron density hangs in balance between the two attractive nuclei; so the density becomes mostly uniform in this middle ground. Hence, the correlation energy approaches the maximum value and is expected to contribute appreciably in the bonding region. Quantitatively, it has been noticed 23 that Local Density Approximation (LDA) theory 24 overestimates the correlation energy roughly by a factor of 2. The LDA is almost identical to the expression in Eq. (1) except that LDA does not have the gradient suppressing factor. In other words, the predicted correlation energy is always at its maximum value for LDA. Since the gradient suppressing factor S(t) ranges from zero to one, a rough estimate would be a factor of 1/2, which would have brought down the LDA s overestimated value roughly to the right one. Finally, we would like to emphasize that the simple meaning of the correlation energy in this work is not an oversimplification, but rather an accurate description of nature, as evident from the results in Fig 1b), Table I, and II. At first glance, it may seem very surprising that one could bypass all the complexities of the perturbation theory, and depend directly on the electron density in order to compute the correlation energy. The fact that this can be achieved as illustrated in Eq. (1) is a result of the fundamental premise of DFT, that the total energy can always be written as a functional of electron density. The true surprise nobody in the 90 year history of quantum mechanics would have suspected, however, is that the functional turns out to be very simple both in the way it can be calculated and interpreted. The correlation energy is the energy that tries to keep electrons in order, for it favors the uniformity of the electrons distribution.

8 8 Acknowledgements Support by Thailand Center of Excellence in Physics grant No. ThEP-60-PET-NU9 is gratefully acknowledged. the Chachiyo theory of electrons in matter Figure 3 Summarizing the Chachiyo theory of electrons in matter. Electron density in the exchange term is specific to the orbital s spin. Method In this work, the Chachiyo theory of electrons in matter as shown in Fig. 3 was implemented in Siam Quantum 25 software package where the exchange and correlation functional were given in Eq. (4) and Eq. (1) respectively. The effect of spin polarization ζ = ρ α ρ β ρ was taken into account via the uniform electron gas correlation energy ε c (r s, ζ) = ε c 0 + (ε c 1 ε c 0 )f(ζ). However, instead of using the vonbarth-hedin weighting function f(ζ) as previously suggested 6, in this work we developed a new weighting function f(ζ) = 2(1 g 3 (ζ)) based on the spin-scaling factor 26 g(ζ) = (1+ζ)2 3 +(1 ζ) in the high density limit. For the exchange energy, the effect of spin polarization was taken into account by computing each spin separately, E x [ρ α, ρ β ] = 1 2 E x[2ρ α ] E x[2ρ β ]. The basis set QZP 27,28 (without G, H orbitals) was used throughout. In Siam Quantum, a basis set was expanded into 6D/10F Cartesian functions; and a numerical grid (75 radial, and 302 Labedev angular) was used in DFT calculations.

9 9 The reported B3LYP, BLYP, PBE, TPSS, and M11 energies were calculated using GAMESS 29. In Fig. 1b, the correlation energies were computed by E (DFT) total E (HF) ; where E (DFT) total was the total energies from the respective DFT methods; and E (HF) was the Hartree-Fock energy calculated under the same basis set. The molecular geometries were taken from the G2 archive 30 which had been optimized by the MP2/6-31G* level of theory. Reference 1. Wigner, E. P. On the interaction of electrons in metals. Phys. Rev. 46, (1934) 2. Szabo, A. & Ostlund, N.S. Modern Quantum Chemistry: Introduction to Advanced Electrical Structure Theory. Dover Publications, Inc. Mineola, New York (1996) 3. Bartlett, R. J. & Musiał, M. Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79, (2007) 4. Møller, C. & Plesset, M. S. Note on an approximation treatment for many-electron systems. Phys. Rev. 46, (1934) 5. Kohn, W. & Sham, L. J. Self-Consisten Equations Including Exchange and Correlation Effects. Phys. Rev. 140, A1133 (1965) 6. Chachiyo, T. Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities. J. Chem. Phys. 145, (2016) 7. Boudreau, J.F. & Swanson, E.S. Applied Computational Physics. Oxford University Press, Oxford (2018)

10 10 8. Fitzgerald, R.J. A simpler ingredient for a complex calculation. Physics Today 69.9, 20 (2016) 9. Ma, S. K. & Brueckner, K. A. Correlation energy of an electron gas with a slowly varying high density. Phys. Rev. 165, (1968) 10. Becke, A. D. Density functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 98, (1993) 11. Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38, (1988) 12. Lee, C., Yang, W. & Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 37, (1988) 13. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, (1996) 14. Tao, J., Perdew, J. P., Staroverov, V. N. & Scuseria, G. E. Climbing the density functional ladder: Nonempirical meta--generalized gradient approximation designed for molecules and solids. Phys. Rev. Lett. 91, (2003) 15. Peverati, R. & Truhlar, D. G. Improving the accuracy of hybrid meta-gga density functionals by range separation. J. Phys. Chem. Lett. 2, (2011) 16. Van Noorden, R., Maher, B. & Nuzzo, R. The top 100 papers. Nature 514, (2014) 17. O neill, D. P. & Gill, P. M. W. Benchmark correlation energies for small molecules. Mol. Phys. 103, (2005) 18. Lide, D. R. CRC Handbook of Chemistry and Physics. CRC Press, Boca Raton, Florida, (2005) 19 Kleinman, L. Exchange density-functional gradient expansion. Phys. Rev. B 30, (1984) 20. March, N. H. Asymptotic formula far from nucleus for exchange energy density in Hartree-Fock theory of closed-shell atoms. Phys. Rev. A 36, (1987)

11 Dirac, P. A. M. Note on Exchange Phenomena in the Thomas Atom. Math. Proc. Cambridge Philos. Soc. 26, (1930) 22. Sun, J., Perdew, J. P. & Ruzsinszky, A. Semilocal density functional obeying a strongly tightened bound for exchange. Proc. Natl. Acad. Sci. 112, (2015) 23. Becke, A. D. Perspective: Fifty years of density-functional theory in chemical physics. J. Chem. Phys. 140, 18A301 (2014) 24. Parr, R.G. & Yang, W. Density-Functional Theory of Atoms and Molecules. Oxford University Press, Oxford (1989) 25. Chachiyo, T. et al. Siam-Quantum: a compact open-source quantum simulation software for molecules, Thailand. (2016) 26. Wang, Y. & Perdew, J. P. Spin scaling of the electron-gas correlation energy in the highdensity limit. Phys. Rev. B 43, (1991) 27. Barbieri, P. L., Fantin, P. A. & Jorge, F. E. Gaussian basis sets of triple and quadruple zeta valence quality for correlated wave functions. Mol. Phys. 104, (2006) 28. Schuchardt, K. L. et al. Basis set exchange: A community database for computational sciences. J. Chem. Inf. Model. 47, (2007) 29. Schmidt, M. W. et al. General atomic and molecular electronic structure system. J. Comput. Chem. 14, (1993) 30. Curtiss, L. A., Raghavachari, K., Trucks, G. W. & Pople, J. A. Gaussian-2 theory for molecular energies of first- and second-row compounds. J. Chem. Phys. 94, (1991)

(mh) 150 100 50 0-50 -100-150 Average Error This Work 5.3 mh B3LYP 5.

12 H He Li Be B C N O F Ne H2 LiH BeH CH CH4 NH NH3 OH H2O FH CO N2 O2 CO2 12 Supplementary Figures Errors of of Calculated Correlation Energies (mhartree) (mh) Average Error This Work 5.3 mh B3LYP 5.3 mh BLYP 18.7 mh PBE 46.4 mh TPSS 34.5 mh M mh This Work B3LYP BLYP PBE TPSS M11

13 Errors of Error Calculated of Total Total Electroic Electronic Energies Energies (mhartree) (mhartree) Average Error This Work 5.0 mh B3LYP 6.4 mh BLYP 20.3 mh PBE 55.4 mh TPSS 40.1 mh M mh H2 LiH BeH CH CH4 NH NH3 OH H2O FH CO N2 O2 CO2 This Work B3LYP BLYP PBE TPSS M Errors of Calculated Ionization Energies (ev) Average Error This Work 0.14 ev B3LYP 0.15 ev BLYP 0.18 ev PBE 0.15 ev TPSS 0.13 ev M ev H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar This Work B3LYP BLYP PBE TPSS M11

Teoría del Funcional de la Densidad (Density Functional Theory)

Teoría del Funcional de la Densidad (Density Functional Theory) Motivation: limitations of the standard approach based on the wave function. The electronic density n(r) as the key variable: Functionals