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Understanding electron correlation energy through density functional theory
Journal Pre-proofs Understanding electron correlation energy through density functional theory Teepanis Chachiyo, Hathaithip Chachiyo PII: DOI: Reference:
S2210-271X(19)30365-2 https://doi.org/10.1016/j.comptc.2019.112669 COMPTC 112669
To appear in:
Computational & Theoretical Chemistry
Received Date: Revised Date: Accepted Date:
26 September 2019 5 December 2019 6 December 2019
Please cite this article as: T. Chachiyo, H. Chachiyo, Understanding electron correlation energy through density functional theory, Computational & Theoretical Chemistry (2019), doi: https://doi.org/10.1016/j.comptc. 2019.112669
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1
Understanding electron correlation energy through density functional theory by Teepanis Chachiyo1,2,* and Hathaithip Chachiyo2
1
Department of Physics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2
Thailand Center of Excellence in Physics, Ministry of Higher Education, Science, Research and
Innovation, 328 Si Ayutthaya Road, Bangkok 10400, Thailand
*
Corresponding author, email: [email protected]
Abstract
A curious behavior of electron correlation energy is explored. Namely, the correlation energy is the energy that tends to drive the system toward that of the uniform electron gas. 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 behavior, 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, the total, the ionization, and the atomization 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.
Keywords: electron correlation energy, generalized gradient approximation, uniform electron gas, density functional theory
2
1. Introduction
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]. Later in the theoretical developments, there are other methods which can calculate the energy very accurately such as the Coupled-Cluster theory [3], Configuration Interaction [2], and Moller-Plesset perturbation theory [4]. In contrast to a perturbation theory where the correlation energy involves a set of molecular orbitals; density functional theory [5] states that the energy depends explicitly on the density of electrons.
Over the past decades, much has been understood regarding the behavior of exchange and correlation energies in density functional theory, especially their asymptotic limits which were used to construct successful functionals such as Becke-88, and PBE. Since the original idea was conceived in the 1920s by Thomas and Fermi [6], progress has been made continuously in the field including the seminal work of Becke [7] in 1988 which brought the error of the exchange energy down to less than 1% (as compared to the exact Hartree-Fock exchange). The Becke-88 functional is also the key ingredient in constructing the B3 hybrid functional [8], the top 10 most cited paper of all time [9]. The equivalently notable success is the non-empirical PBE exchange and correlation functionals [10] whose parameters satisfy 11 exact constraints [11]. An even larger set of criteria, 17 conditions, is satisfied by the SCAN functional [12] which belongs to the state-of-the-art meta-GGA group. The meta-GGA functionals [13] use the electron density 𝜌(𝑟), its gradient ∇𝜌(𝑟), and the positive orbital 1
2
kinetic energy density ∑𝑛2|∇𝜓𝑛| to evaluate the exchange and correlation energy. Continuous advances are also being made in the high temperature conditions [14].
However, in addition to the asymptotic limits, the behavior in the intermediate region should be as equally important. As such, two functionals obeying similar limits could give different predictions if they behave differently in the region connecting those limits. In this study, we explore alternative ideas that lead to simple and accurate correlation functional. Tests on atoms and molecules are presented. We further discuss how the functional is related to the conditions obeyed by the PBE functional.
3
To gain an initial insight into how electron correlation affects electron density, we first consider an exactly solvable model where there is no need to approximate the solution neither through a series of perturbations, nor a set of predefined basis functions. Shown in Fig. 1(a) is the “Hooke atom” [15], 1
1
where the two electrons are bound to the origin by a harmonic potential 𝑉(𝑟) = 2𝑘𝑟2. For 𝑘 = 4, the Schrödinger equation was solved exactly in 1990s by Kais et al. yielding the ground state total energy and the ground state wave function [16].
Effects of Electron Correlation on Hooke Atom (a) Exactly Solvable Model
(b) Electron Densities 0.1
Without Correlation
0.09
Coulomb Repulsion
0.08
Include Correlation Exactly
0.07
Harmonic Potential
0.06
0.25
0.5
0.75
1
(d) Change of Gradient
(c) Change of Electron Density 0.001
0
0.001
0 0
2
4
2
4
6
6 0.001
0.001
0.002
0.002
0.003
Fig. 1. (a) Hooke atom. (b) Electron densities. The Hartree-Fock calculation is reproduced as described by Ref. [16]. (c) Change of electron density due to correlation: 𝜌exact ― 𝜌HF. (d) Change of gradient due to correlation: |∇𝜌|exact ― |∇𝜌|HF. The horizontal axis indicates the distance from the center of the Hooke atom.
4
Figure 1(a) illustrates the two electrons in the Hooke atom and the exact wave function. Figure 1(b) is essentially the same as in the previous study [16], zooming into the region where the difference of electron density is clearly visible. A broader view of the difference, 𝜌exact ― 𝜌HF , is shown in Fig. 1(c). Finally, the gradient of both electron densities are compared in Fig. 1(d).
Looking closely in Fig. 1(b), one observes that the exact density changes less rapidly than the Hartree-Fock density. Since the Hartree-Fock method contains no correlation; it must be the effect of correlation in the exact density that drives the system closer to homogeneity. The change of gradient in Fig. 1(d) is also predominantly negative, meaning the overall gradient of the system is reduced due to electron correlation. In this specific example, the exactly solvable solution shows that electron correlation favors more homogeneous distribution of electrons.
Is it possible that the tendency to drive the system toward homogeneity is also a characteristic of electron correlation in general, applicable to all atoms and molecules? In this work, we entertain such hypothesis from which a mathematical correlation functional can be derived. The accuracy of the functional will then hint our understanding of the electron correlation energy, and how it affects the density of electrons in general. In spite of the tendency, the condition in which electron correlation may not succeed in driving the system toward homogeneity is also discussed.
2. Methods
We hypothesize that the correlation energy is the energy that tends to drive the system toward the uniform electron gas. The hypothesis leads to a very simple expression as follows.
ℎ
𝐸𝑐 = ∫𝜌𝜀𝑐(1 + 𝑡2)
𝜀𝑐
𝑑3𝑟
(1)
5
Here, 𝜌 is the total electron density; and 𝜀𝑐 is the correlation energy density when the density is 𝑏
𝑏
perfectly uniform, which we use the Chachiyo’s formula [17-19] 𝜀𝑐 = 𝑎ln (1 + 𝑟𝑠 + 𝑟2). The term 𝑠
ℎ 2
(1 + 𝑡 )
𝜀𝑐
is the gradient suppressing factor, designed to attenuate the energy from its maximum
value as a gradient parameter [20] 𝑡 = 3 13
(8.470 × 10 ―3) × 16(𝜋)
1 6 𝑎0|∇𝜌|
(𝜋3)
4 𝜌7 6
1
increases. In addition, the constant ℎ = 2
= 0.06672632 Hartree is related to the behavior of the correlation
energy when electron density varies very slowly.
(a)
Maximum Suppressing factor gradient
(b)
CO2
O2
N2
CO
FH
H2O
OH
NH3
NH
CH4
CH
BeH
LiH
H2
Ne
F
O
N
C
B
Be
Li
He
0
H
Calculated Correlation Energies (mHartree) Calculated Correlation Energies (mHartree) using correlation functional
-100 -200 -300 -400 -500 -600
Experiment
-700 -800
This Work
(Ref.28)
MAE (allow error cancellation) 5.3 mH (no error cancellation) 31.0 mH
-900 Experiment
This Work
No Cancellation
Fig. 2. (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 correlation energies are computed in two ways: (i) allowing (𝐻𝐹) (𝐷𝐹𝑇) error cancellation 𝐸(𝐷𝐹𝑇) . total ― 𝐸total , and (ii) without error cancellation 𝐸𝑐
The derivation of Eq. (1) is simple but not rigorous. As shown in Fig. 2(a) 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 initially accounted for by using a “gradient suppressing factor” 𝑆(𝑡) in a form of Gaussian decay.
6
Initial Form:
2
𝑆(𝑡) = 𝑒 ―𝛽𝑡
(2)
It was exactly the shape of a bell curve, monotonically decaying as 𝑡 increased. It was also consistent with the slowly varying density limit in which Ma and Brueckner [21] derived that
[
𝐸𝑐→∫ 𝜌𝜀𝑐 + 8.470 × 10
|∇𝜌|2
―3
𝜌
43
]𝑑 𝑟 3
as |∇𝜌|→0
Rydberg
(3)
2
As the gradient approached zero, the 𝑒 ―𝛽𝑡 →(1 ― 𝛽𝑡2). Therefore, in this limit
[
where the factor
𝜋 131 3 16
()
2
3
2 𝜋 1 3 1 |∇𝜌| 3 16 𝜌4 3
()
∫𝜌𝜀𝑐𝑆(𝑡) 𝑑 𝑟→∫𝜌𝜀𝑐[1 ― 𝛽𝑡 ] 𝑑 𝑟 = ∫ 𝜌𝜀𝑐 ― 𝜀𝑐𝛽 3
] 𝑑 𝑟, 3
(4)
came from the definition of 𝑡. Demanding that Eq. (3) and Eq. (4) agreed,
we had
3 13 𝜋
()
1
𝛽 = ― ℎ 𝜀𝑐, and ℎ = 2(8.470 × 10 ―3) × 16
= 0.06672632 Hartree
(5)
2
However, the 𝑡2 dependence in the initial form 𝑆(𝑡) = 𝑒 ―𝛽𝑡 was not appropriate when the gradient became larger. As pointed out by Perdew, Burke, and Ernzerhof when they constructed the highly successful non-empirical PBE [10] functional, the dependence ought to be proportional to ln 𝑡2 in order to cancel the logarithmic divergence of the uniform electron gas correlation energy.
In this work, we reconciled the two domains by using ln (1 + 𝑡2) instead. That way, it converged to 𝑡2 when 𝑡 was small, and approached ln 𝑡2 when 𝑡 was large. Therefore, we arrived at the
7
Final Form:
𝑆(𝑡) = 𝑒
―𝛽ln (1 + 𝑡2)
=𝑒
ℎ 𝜀𝑐ln
(1 + 𝑡2)
,
(6)
which promptly led to the correlation functional in Eq. (1).
In the presence of a spin polarization 𝜁 =
𝜌𝛼 ― 𝜌𝛽 𝜌
, the uniform electron gas correlation energy density
𝜀𝑐 could be written as an interpolation between two extreme cases: the paramagnetic and the ferromagnetic.
𝜀𝑐(𝑟𝑠,𝜁) = 𝜀0𝑐 + (𝜀1𝑐 ― 𝜀0𝑐 )𝑓(𝜁)
(7)
1
In the original paper [17], judging from the similar 𝑟𝑠 behavior in the low density limit, we had suggested the vonBarth-Hedin’s 𝑓vBH(𝜁) =
43
(1 + 𝜁)
43
+ (1 ― 𝜁) 13
2(2
―2
― 1)
initially developed for the exchange
energy [22] be used as the weighting function for the correlation energy density as well.
In 1991 Wang and Perdew [23] had studied how the spin polarization affected the correlation energy density 𝜀𝑐 in the high density limit. They found that the vBH was not as accurate, and proposed a spin scaling factor 𝐼Wang ― Perdew = 𝑔3(𝜁) where 𝑔(𝜁) =
23
(1 + 𝜁)
23
+ (1 ― 𝜁) 2
.
However, Wang and Perdew casted their formalism in terms of a scaling factor; but what we needed was a better weighting function. As such, we turned the scaling factor 𝑔3(𝜁) into a weighting function 𝑓(𝜁) in the same way (but in reverse) they turned vBH weighting function into a factor (𝐼vBH 1
= 1 ― 2𝑓vBH(𝜁), see Eq. 43 in Wang and Perdew, 1991).
Therefore, in this work we propose using
8
𝑓(𝜁) = 2(1 ― 𝑔3(𝜁))
(8)
as a weighting function when taking into account of a spin polarization in a system.
To produce the results as shown in the following section one also needs an exchange energy contribution which we used our previously proposed exchange functional [24].
3𝑥2 + 𝜋2ln (𝑥 + 1)
𝐸𝑥 = ∫𝜌𝜀𝑥(3𝑥 + 𝜋2)ln (𝑥 + 1)𝑑3𝑟.
(9)
Here, 𝜀𝑥 is the Dirac exchange energy density [25] for uniform electron gas; and the parameter 𝑥 = |∇𝜌|2 𝜋 1 3 𝜌
(3)
4 39
is used to represent the gradient of the electron density.
Recently, the functional was also implemented in the LibXC [26] under the name “Chachiyo exchange”. It was also used to compute electron densities in a newly proposed method for building initial guesses called SAP (Superposition of Atomic Potential) [27]. The study reported that, out of hundreds functionals available in LibXC, the Chachiyo exchange yielded the best initial guess wave functions on average.
3. Results and discussions
9
(a)
Error ofofCalculated Total Energies Error Total Electroic Energies (mH) (mHartree)
150 100
Average |Error| This Work 5.0 mH B3LYP 6.4 mH 20.3 mH BLYP PBE 55.4 mH TPSS 40.1 mH M11 9.6 mH
50 0 -50 -100 -150 H2
LiH BeH CH CH4 NH NH3 OH H2O FH
This Work
150
BLYP
PBE
N2
TPSS
O2 CO2 M11
ErrorofofCalculated Calculated Correlation (mHartree) Errors Correlation EnergiesEnergies (mH) H He Li Be B C N O F Ne H2 LiH BeH CH CH4 NH NH3 OH H2O FH CO N2 O2 CO2
(b)
B3LYP
CO
Average |Error| This Work 5.3 mH B3LYP 5.3 mH 18.7 mH BLYP PBE 46.4 mH TPSS 34.5 mH M11 11.6 mH
100 50 0 -50 -100 -150 This Work
B3LYP
BLYP
PBE
TPSS
M11
Fig. 3. Error of calculated energies: (a) total energies, and (b) correlation energies. The exact energies are from Ref. [28].
10
(a)
Error of Calculated Ionization Energies (mHartree)
Errors of Calculated Ionization Energies (mH)
25 20 15 10 5 0 -5
H He Li
Be
B
C
N
O
F
Ne Na Mg Al
Si
P
S
Cl
Ar
-10 -15
Average |Error| This Work 5.1 mH B3LYP 5.4 mH BLYP 6.7 mH PBE 5.5 mH TPSS 4.9 mH M11 5.2 mH
-20 This Work
B3LYP
BLYP
PBE
TPSS
M11
Errorofof Calculated Atomization Energies (mHartree) (b) Errors Calculated Atomization Energies (mH) 50 40 30 20 10 0 -10
H2
LiH BeH CH CH4 NH NH3 OH OH2 FH
CO
N2
O2 CO2
Average |Error| This Work 6.4 mH B3LYP 4.3 mH BLYP 7.4 mH PBE 11.7 mH TPSS 6.0 mH
-20 This Work
B3LYP
BLYP
PBE
TPSS
Fig. 4. Error of calculated energies: (a) ionization energies, and (b) atomization energies. The predicted ionization energies are computed using Delta-SCF method. The experimental values are from Ref. [29]. Zero point energies and relativistic corrections have been subtracted from the experimental atomization values [28].
Fig. 2(b), Fig. 3, and Fig. 4 illustrate the accuracy of the correlation functional in Eq. (1) via the calculated correlation, the total, the ionization, and the atomization energies of atoms and molecules. The performance is comparable to that of B3LYP [8], BLYP [7,30], PBE [10], TPSS [13], and M11 [31]. Considering that the five functionals are among the most cited works of all time [9], the results validate the accuracy of the correlation energy functional in Eq. (1).
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.
11
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 [32] that Local Density Approximation (LDA) theory [6] 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.
We now discuss the results in Fig. 2(b) where the correlation energies in this work are computed in (𝐻𝐹) (𝐷𝐹𝑇) two different ways: (i) 𝐸(𝐷𝐹𝑇) . The former puts more emphasis on the total ― 𝐸total , and (ii) 𝐸𝑐
performance of each method when predicting the total energy. The reported correlation energy is a reflection of this performance. The errors of the exchange and the correlation are allowed to cancel (i.e. cancellation of errors). The latter, however, does not allow error cancellation. The results indicate large degree of error cancellation which reduces the mean absolute error from 31.0 mH to 5.3 mH. Therefore, the exchange and correlation functional in this work perform better when used together as a combination.
Fig. 3 and Fig. 4 show the predictions from B3LYP, BLYP, PBE, M11, TPSS, and the functional in this work. Ideally, the error shown in Fig. 3(a) and Fig. 3(b) should have been identical if our calculated Hartree-Fock energies and that of the reference paper [28] were identical. Instead they are off by a few milli-hartrees because the reference paper used an extremely large basis set: cc-pV5Z-h. Nevertheless, the presented data is sufficient to conclude that our functional’s accuracy is comparable to that of the other methods.
12
In contrast to the total energies, the error of predicted ionization energies and atomization energies are more similar among the various functionals shown in Fig. 4. Because the sample size is small, the data is not enough to rank the performance of the functionals, only enough to indicate that our functionals are competitive to the others.
Instead of the average errors, the differences are more pronounced if one focuses on the systematic errors, where various functionals consistently overestimate (or underestimate) the correlation and the total energies. We attribute these differences to the intermediate region connecting the asymptotic limits, and how the functionals are constructed to satisfy those limits.
0.06 Higher Density
0.04
Lower Density
0.02
0
2
4
6
8
Fig. 5. Comparison between the 𝐻(𝑟𝑠,𝑡) from the PBE correlation and [𝑆(𝑟𝑠,𝑡) ― 1]𝜀𝑐(𝑟𝑠) for two different electron densities.
The correlation functional in this work follows the three conditions used in the PBE correlation. The difference is in the way we mathematically construct our functional. PBE uses the function 𝐻(𝑟𝑠,𝑡) to add to the uniform electron gas correlation energy, while we propose a function 𝑆(𝑟𝑠,𝑡) that multiplies to it instead. The mathematical forms are also different. Figure 5 shows how the two functionals respond to the gradient parameter 𝑡. Note how the two functionals obey the same asymptotic conditions, but behave differently in the intermediate region, especially when the electron
13
density is higher. Therefore, we attribute the success of our correlation functional to its behavior in the intermediate region.
The tendency toward slowly varying electron gas, however, is not the only preferential behavior due to electron correlation. Because the strength of uniform electron gas correlation energy density 𝜀𝑐(𝑟𝑠) increases as a function of electron density; the system also prefers higher density. Since the total number of electrons is conserved via normalization, accumulating higher density in one region implies lower density in the adjacent areas, promoting higher gradient of electron density. Therefore, the tendency toward higher density may sometimes compete with the tendency toward homogeneity.
Most recently, it is shown [33] that for the large-Z limit of neutral atoms, electron correlation becomes local and slowly-varying. Further study is needed to address how the functional in this work performs in such limit, along with other important cases such as van der Waals interaction, hydrogen bonding, or bulk crystal.
4. Conclusions
Finally, we would like to emphasize that the simple behavior 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. 2(b), Fig. 3, and Fig. 4. 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 [34,10]. The true surprise nobody in the 90 year history of quantum mechanics would have suspected, however, is that the functional may be very simple both in the way it can be calculated and interpreted. We propose that the correlation energy is the energy that locally tends to drive the system toward the uniform electron gas.
14
Supplementary Material See supplementary material for additional methods for spin polarized cases, computational details, and computer source codes.
Acknowledgements Support by Thailand Center of Excellence in Physics grant No. ThEP-60-PET-NU9 is gratefully acknowledged.
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[21] S. Ma, K.A. Brueckner, Correlation Energy of an Electron Gas with a Slowly Varying High Density, Phys. Rev. 165 (1968) 18–31. doi:10.1103/PhysRev.165.18. [22] U. Von Barth, L. Hedin, A local exchange-correlation potential for the spin polarized case. I, J. Phys. C Solid State Phys. 5 (1972) 1629–1642. doi:10.1088/0022-3719/5/13/012. [23] Y. Wang, J.P. Perdew, Spin scaling of the electron-gas correlation energy in the high-density limit, Phys. Rev. B. 43 (1991) 8911–8916. doi:10.1103/PhysRevB.43.8911. [24] T. Chachiyo, H. Chachiyo, Simple and accurate exchange energy for density functional theory, ArXiv Prepr. ArXiv1706.01343. (2017). http://arxiv.org/abs/1706.01343. [25] P.A.M. Dirac, Note on Exchange Phenomena in the Thomas Atom, Math. Proc. Cambridge Philos. Soc. 26 (1930) 376–385. doi:10.1017/S0305004100016108. [26] S. Lehtola, C. Steigemann, M.J.T. Oliveira, M.A.L. Marques, Recent developments in LIBXC — A comprehensive library of functionals for density functional theory, SoftwareX. 7 (2018) 1–5. doi:10.1016/j.softx.2017.11.002. [27] S. Lehtola, Assessment of Initial Guesses for Self-Consistent Field Calculations. Superposition of Atomic Potentials: Simple yet Efficient, J. Chem. Theory Comput. 15 (2019) 1593–1604. doi:10.1021/acs.jctc.8b01089. [28] D.P. O’neill, P.M.W. Gill, Benchmark correlation energies for small molecules, Mol. Phys. 103 (2005) 763–766. doi:10.1080/00268970512331339323. [29] D.R. Lide, CRC Handbook of Chemistry and Physics: A Ready-Reference of Chemical and Physical Data, CRC press, Florida, 2005. doi:10.1021/ja041017a. [30] C. Lee, W. Yang, R.G. Parr, Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density, Phys. Rev. B. 37 (1988) 785–789. doi:10.1103/PhysRevB.37.785. [31] R. Peverati, D.G. Truhlar, Improving the accuracy of hybrid meta-GGA density functionals by range separation, J. Phys. Chem. Lett. 2 (2011) 2810–2817. doi:10.1021/jz201170d. [32] A.D. Becke, Perspective: Fifty years of density-functional theory in chemical physics, J. Chem. Phys. 140 (2014) 18A301. doi:10.1063/1.4869598. [33] K. Burke, A. Cancio, T. Gould, S. Pittalis, Locality of correlation in density functional theory, J. Chem. Phys. 145 (2016) 054112. doi:10.1063/1.4959126.
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[34] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964) B864–B871. doi:10.1103/PhysRev.136.B864.
Teepanis Chachiyo: Conceptualization, Methodology, Software, Writing- Original draft Hathaithip Chachiyo: Data curation, Validation, Writing- Reviewing and Editing,
18
A very simple way to compute electron correlation energy Reasonably accurate for covalent bond molecules, competitive to B3LYP Hypothesis: the correlation energy favors uniform electron distribution
S2210-271X(19)30365-2 https://doi.org/10.1016/j.comptc.2019.112669 COMPTC 112669
To appear in:
Computational & Theoretical Chemistry
Received Date: Revised Date: Accepted Date:
26 September 2019 5 December 2019 6 December 2019
Please cite this article as: T. Chachiyo, H. Chachiyo, Understanding electron correlation energy through density functional theory, Computational & Theoretical Chemistry (2019), doi: https://doi.org/10.1016/j.comptc. 2019.112669
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1
Understanding electron correlation energy through density functional theory by Teepanis Chachiyo1,2,* and Hathaithip Chachiyo2
1
Department of Physics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2
Thailand Center of Excellence in Physics, Ministry of Higher Education, Science, Research and
Innovation, 328 Si Ayutthaya Road, Bangkok 10400, Thailand
*
Corresponding author, email: [email protected]
Abstract
A curious behavior of electron correlation energy is explored. Namely, the correlation energy is the energy that tends to drive the system toward that of the uniform electron gas. 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 behavior, 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, the total, the ionization, and the atomization 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.
Keywords: electron correlation energy, generalized gradient approximation, uniform electron gas, density functional theory
2
1. Introduction
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]. Later in the theoretical developments, there are other methods which can calculate the energy very accurately such as the Coupled-Cluster theory [3], Configuration Interaction [2], and Moller-Plesset perturbation theory [4]. In contrast to a perturbation theory where the correlation energy involves a set of molecular orbitals; density functional theory [5] states that the energy depends explicitly on the density of electrons.
Over the past decades, much has been understood regarding the behavior of exchange and correlation energies in density functional theory, especially their asymptotic limits which were used to construct successful functionals such as Becke-88, and PBE. Since the original idea was conceived in the 1920s by Thomas and Fermi [6], progress has been made continuously in the field including the seminal work of Becke [7] in 1988 which brought the error of the exchange energy down to less than 1% (as compared to the exact Hartree-Fock exchange). The Becke-88 functional is also the key ingredient in constructing the B3 hybrid functional [8], the top 10 most cited paper of all time [9]. The equivalently notable success is the non-empirical PBE exchange and correlation functionals [10] whose parameters satisfy 11 exact constraints [11]. An even larger set of criteria, 17 conditions, is satisfied by the SCAN functional [12] which belongs to the state-of-the-art meta-GGA group. The meta-GGA functionals [13] use the electron density 𝜌(𝑟), its gradient ∇𝜌(𝑟), and the positive orbital 1
2
kinetic energy density ∑𝑛2|∇𝜓𝑛| to evaluate the exchange and correlation energy. Continuous advances are also being made in the high temperature conditions [14].
However, in addition to the asymptotic limits, the behavior in the intermediate region should be as equally important. As such, two functionals obeying similar limits could give different predictions if they behave differently in the region connecting those limits. In this study, we explore alternative ideas that lead to simple and accurate correlation functional. Tests on atoms and molecules are presented. We further discuss how the functional is related to the conditions obeyed by the PBE functional.
3
To gain an initial insight into how electron correlation affects electron density, we first consider an exactly solvable model where there is no need to approximate the solution neither through a series of perturbations, nor a set of predefined basis functions. Shown in Fig. 1(a) is the “Hooke atom” [15], 1
1
where the two electrons are bound to the origin by a harmonic potential 𝑉(𝑟) = 2𝑘𝑟2. For 𝑘 = 4, the Schrödinger equation was solved exactly in 1990s by Kais et al. yielding the ground state total energy and the ground state wave function [16].
Effects of Electron Correlation on Hooke Atom (a) Exactly Solvable Model
(b) Electron Densities 0.1
Without Correlation
0.09
Coulomb Repulsion
0.08
Include Correlation Exactly
0.07
Harmonic Potential
0.06
0.25
0.5
0.75
1
(d) Change of Gradient
(c) Change of Electron Density 0.001
0
0.001
0 0
2
4
2
4
6
6 0.001
0.001
0.002
0.002
0.003
Fig. 1. (a) Hooke atom. (b) Electron densities. The Hartree-Fock calculation is reproduced as described by Ref. [16]. (c) Change of electron density due to correlation: 𝜌exact ― 𝜌HF. (d) Change of gradient due to correlation: |∇𝜌|exact ― |∇𝜌|HF. The horizontal axis indicates the distance from the center of the Hooke atom.
4
Figure 1(a) illustrates the two electrons in the Hooke atom and the exact wave function. Figure 1(b) is essentially the same as in the previous study [16], zooming into the region where the difference of electron density is clearly visible. A broader view of the difference, 𝜌exact ― 𝜌HF , is shown in Fig. 1(c). Finally, the gradient of both electron densities are compared in Fig. 1(d).
Looking closely in Fig. 1(b), one observes that the exact density changes less rapidly than the Hartree-Fock density. Since the Hartree-Fock method contains no correlation; it must be the effect of correlation in the exact density that drives the system closer to homogeneity. The change of gradient in Fig. 1(d) is also predominantly negative, meaning the overall gradient of the system is reduced due to electron correlation. In this specific example, the exactly solvable solution shows that electron correlation favors more homogeneous distribution of electrons.
Is it possible that the tendency to drive the system toward homogeneity is also a characteristic of electron correlation in general, applicable to all atoms and molecules? In this work, we entertain such hypothesis from which a mathematical correlation functional can be derived. The accuracy of the functional will then hint our understanding of the electron correlation energy, and how it affects the density of electrons in general. In spite of the tendency, the condition in which electron correlation may not succeed in driving the system toward homogeneity is also discussed.
2. Methods
We hypothesize that the correlation energy is the energy that tends to drive the system toward the uniform electron gas. The hypothesis leads to a very simple expression as follows.
ℎ
𝐸𝑐 = ∫𝜌𝜀𝑐(1 + 𝑡2)
𝜀𝑐
𝑑3𝑟
(1)
5
Here, 𝜌 is the total electron density; and 𝜀𝑐 is the correlation energy density when the density is 𝑏
𝑏
perfectly uniform, which we use the Chachiyo’s formula [17-19] 𝜀𝑐 = 𝑎ln (1 + 𝑟𝑠 + 𝑟2). The term 𝑠
ℎ 2
(1 + 𝑡 )
𝜀𝑐
is the gradient suppressing factor, designed to attenuate the energy from its maximum
value as a gradient parameter [20] 𝑡 = 3 13
(8.470 × 10 ―3) × 16(𝜋)
1 6 𝑎0|∇𝜌|
(𝜋3)
4 𝜌7 6
1
increases. In addition, the constant ℎ = 2
= 0.06672632 Hartree is related to the behavior of the correlation
energy when electron density varies very slowly.
(a)
Maximum Suppressing factor gradient
(b)
CO2
O2
N2
CO
FH
H2O
OH
NH3
NH
CH4
CH
BeH
LiH
H2
Ne
F
O
N
C
B
Be
Li
He
0
H
Calculated Correlation Energies (mHartree) Calculated Correlation Energies (mHartree) using correlation functional
-100 -200 -300 -400 -500 -600
Experiment
-700 -800
This Work
(Ref.28)
MAE (allow error cancellation) 5.3 mH (no error cancellation) 31.0 mH
-900 Experiment
This Work
No Cancellation
Fig. 2. (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 correlation energies are computed in two ways: (i) allowing (𝐻𝐹) (𝐷𝐹𝑇) error cancellation 𝐸(𝐷𝐹𝑇) . total ― 𝐸total , and (ii) without error cancellation 𝐸𝑐
The derivation of Eq. (1) is simple but not rigorous. As shown in Fig. 2(a) 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 initially accounted for by using a “gradient suppressing factor” 𝑆(𝑡) in a form of Gaussian decay.
6
Initial Form:
2
𝑆(𝑡) = 𝑒 ―𝛽𝑡
(2)
It was exactly the shape of a bell curve, monotonically decaying as 𝑡 increased. It was also consistent with the slowly varying density limit in which Ma and Brueckner [21] derived that
[
𝐸𝑐→∫ 𝜌𝜀𝑐 + 8.470 × 10
|∇𝜌|2
―3
𝜌
43
]𝑑 𝑟 3
as |∇𝜌|→0
Rydberg
(3)
2
As the gradient approached zero, the 𝑒 ―𝛽𝑡 →(1 ― 𝛽𝑡2). Therefore, in this limit
[
where the factor
𝜋 131 3 16
()
2
3
2 𝜋 1 3 1 |∇𝜌| 3 16 𝜌4 3
()
∫𝜌𝜀𝑐𝑆(𝑡) 𝑑 𝑟→∫𝜌𝜀𝑐[1 ― 𝛽𝑡 ] 𝑑 𝑟 = ∫ 𝜌𝜀𝑐 ― 𝜀𝑐𝛽 3
] 𝑑 𝑟, 3
(4)
came from the definition of 𝑡. Demanding that Eq. (3) and Eq. (4) agreed,
we had
3 13 𝜋
()
1
𝛽 = ― ℎ 𝜀𝑐, and ℎ = 2(8.470 × 10 ―3) × 16
= 0.06672632 Hartree
(5)
2
However, the 𝑡2 dependence in the initial form 𝑆(𝑡) = 𝑒 ―𝛽𝑡 was not appropriate when the gradient became larger. As pointed out by Perdew, Burke, and Ernzerhof when they constructed the highly successful non-empirical PBE [10] functional, the dependence ought to be proportional to ln 𝑡2 in order to cancel the logarithmic divergence of the uniform electron gas correlation energy.
In this work, we reconciled the two domains by using ln (1 + 𝑡2) instead. That way, it converged to 𝑡2 when 𝑡 was small, and approached ln 𝑡2 when 𝑡 was large. Therefore, we arrived at the
7
Final Form:
𝑆(𝑡) = 𝑒
―𝛽ln (1 + 𝑡2)
=𝑒
ℎ 𝜀𝑐ln
(1 + 𝑡2)
,
(6)
which promptly led to the correlation functional in Eq. (1).
In the presence of a spin polarization 𝜁 =
𝜌𝛼 ― 𝜌𝛽 𝜌
, the uniform electron gas correlation energy density
𝜀𝑐 could be written as an interpolation between two extreme cases: the paramagnetic and the ferromagnetic.
𝜀𝑐(𝑟𝑠,𝜁) = 𝜀0𝑐 + (𝜀1𝑐 ― 𝜀0𝑐 )𝑓(𝜁)
(7)
1
In the original paper [17], judging from the similar 𝑟𝑠 behavior in the low density limit, we had suggested the vonBarth-Hedin’s 𝑓vBH(𝜁) =
43
(1 + 𝜁)
43
+ (1 ― 𝜁) 13
2(2
―2
― 1)
initially developed for the exchange
energy [22] be used as the weighting function for the correlation energy density as well.
In 1991 Wang and Perdew [23] had studied how the spin polarization affected the correlation energy density 𝜀𝑐 in the high density limit. They found that the vBH was not as accurate, and proposed a spin scaling factor 𝐼Wang ― Perdew = 𝑔3(𝜁) where 𝑔(𝜁) =
23
(1 + 𝜁)
23
+ (1 ― 𝜁) 2
.
However, Wang and Perdew casted their formalism in terms of a scaling factor; but what we needed was a better weighting function. As such, we turned the scaling factor 𝑔3(𝜁) into a weighting function 𝑓(𝜁) in the same way (but in reverse) they turned vBH weighting function into a factor (𝐼vBH 1
= 1 ― 2𝑓vBH(𝜁), see Eq. 43 in Wang and Perdew, 1991).
Therefore, in this work we propose using
8
𝑓(𝜁) = 2(1 ― 𝑔3(𝜁))
(8)
as a weighting function when taking into account of a spin polarization in a system.
To produce the results as shown in the following section one also needs an exchange energy contribution which we used our previously proposed exchange functional [24].
3𝑥2 + 𝜋2ln (𝑥 + 1)
𝐸𝑥 = ∫𝜌𝜀𝑥(3𝑥 + 𝜋2)ln (𝑥 + 1)𝑑3𝑟.
(9)
Here, 𝜀𝑥 is the Dirac exchange energy density [25] for uniform electron gas; and the parameter 𝑥 = |∇𝜌|2 𝜋 1 3 𝜌
(3)
4 39
is used to represent the gradient of the electron density.
Recently, the functional was also implemented in the LibXC [26] under the name “Chachiyo exchange”. It was also used to compute electron densities in a newly proposed method for building initial guesses called SAP (Superposition of Atomic Potential) [27]. The study reported that, out of hundreds functionals available in LibXC, the Chachiyo exchange yielded the best initial guess wave functions on average.
3. Results and discussions
9
(a)
Error ofofCalculated Total Energies Error Total Electroic Energies (mH) (mHartree)
150 100
Average |Error| This Work 5.0 mH B3LYP 6.4 mH 20.3 mH BLYP PBE 55.4 mH TPSS 40.1 mH M11 9.6 mH
50 0 -50 -100 -150 H2
LiH BeH CH CH4 NH NH3 OH H2O FH
This Work
150
BLYP
PBE
N2
TPSS
O2 CO2 M11
ErrorofofCalculated Calculated Correlation (mHartree) Errors Correlation EnergiesEnergies (mH) H He Li Be B C N O F Ne H2 LiH BeH CH CH4 NH NH3 OH H2O FH CO N2 O2 CO2
(b)
B3LYP
CO
Average |Error| This Work 5.3 mH B3LYP 5.3 mH 18.7 mH BLYP PBE 46.4 mH TPSS 34.5 mH M11 11.6 mH
100 50 0 -50 -100 -150 This Work
B3LYP
BLYP
PBE
TPSS
M11
Fig. 3. Error of calculated energies: (a) total energies, and (b) correlation energies. The exact energies are from Ref. [28].
10
(a)
Error of Calculated Ionization Energies (mHartree)
Errors of Calculated Ionization Energies (mH)
25 20 15 10 5 0 -5
H He Li
Be
B
C
N
O
F
Ne Na Mg Al
Si
P
S
Cl
Ar
-10 -15
Average |Error| This Work 5.1 mH B3LYP 5.4 mH BLYP 6.7 mH PBE 5.5 mH TPSS 4.9 mH M11 5.2 mH
-20 This Work
B3LYP
BLYP
PBE
TPSS
M11
Errorofof Calculated Atomization Energies (mHartree) (b) Errors Calculated Atomization Energies (mH) 50 40 30 20 10 0 -10
H2
LiH BeH CH CH4 NH NH3 OH OH2 FH
CO
N2
O2 CO2
Average |Error| This Work 6.4 mH B3LYP 4.3 mH BLYP 7.4 mH PBE 11.7 mH TPSS 6.0 mH
-20 This Work
B3LYP
BLYP
PBE
TPSS
Fig. 4. Error of calculated energies: (a) ionization energies, and (b) atomization energies. The predicted ionization energies are computed using Delta-SCF method. The experimental values are from Ref. [29]. Zero point energies and relativistic corrections have been subtracted from the experimental atomization values [28].
Fig. 2(b), Fig. 3, and Fig. 4 illustrate the accuracy of the correlation functional in Eq. (1) via the calculated correlation, the total, the ionization, and the atomization energies of atoms and molecules. The performance is comparable to that of B3LYP [8], BLYP [7,30], PBE [10], TPSS [13], and M11 [31]. Considering that the five functionals are among the most cited works of all time [9], the results validate the accuracy of the correlation energy functional in Eq. (1).
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.
11
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 [32] that Local Density Approximation (LDA) theory [6] 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.
We now discuss the results in Fig. 2(b) where the correlation energies in this work are computed in (𝐻𝐹) (𝐷𝐹𝑇) two different ways: (i) 𝐸(𝐷𝐹𝑇) . The former puts more emphasis on the total ― 𝐸total , and (ii) 𝐸𝑐
performance of each method when predicting the total energy. The reported correlation energy is a reflection of this performance. The errors of the exchange and the correlation are allowed to cancel (i.e. cancellation of errors). The latter, however, does not allow error cancellation. The results indicate large degree of error cancellation which reduces the mean absolute error from 31.0 mH to 5.3 mH. Therefore, the exchange and correlation functional in this work perform better when used together as a combination.
Fig. 3 and Fig. 4 show the predictions from B3LYP, BLYP, PBE, M11, TPSS, and the functional in this work. Ideally, the error shown in Fig. 3(a) and Fig. 3(b) should have been identical if our calculated Hartree-Fock energies and that of the reference paper [28] were identical. Instead they are off by a few milli-hartrees because the reference paper used an extremely large basis set: cc-pV5Z-h. Nevertheless, the presented data is sufficient to conclude that our functional’s accuracy is comparable to that of the other methods.
12
In contrast to the total energies, the error of predicted ionization energies and atomization energies are more similar among the various functionals shown in Fig. 4. Because the sample size is small, the data is not enough to rank the performance of the functionals, only enough to indicate that our functionals are competitive to the others.
Instead of the average errors, the differences are more pronounced if one focuses on the systematic errors, where various functionals consistently overestimate (or underestimate) the correlation and the total energies. We attribute these differences to the intermediate region connecting the asymptotic limits, and how the functionals are constructed to satisfy those limits.
0.06 Higher Density
0.04
Lower Density
0.02
0
2
4
6
8
Fig. 5. Comparison between the 𝐻(𝑟𝑠,𝑡) from the PBE correlation and [𝑆(𝑟𝑠,𝑡) ― 1]𝜀𝑐(𝑟𝑠) for two different electron densities.
The correlation functional in this work follows the three conditions used in the PBE correlation. The difference is in the way we mathematically construct our functional. PBE uses the function 𝐻(𝑟𝑠,𝑡) to add to the uniform electron gas correlation energy, while we propose a function 𝑆(𝑟𝑠,𝑡) that multiplies to it instead. The mathematical forms are also different. Figure 5 shows how the two functionals respond to the gradient parameter 𝑡. Note how the two functionals obey the same asymptotic conditions, but behave differently in the intermediate region, especially when the electron
13
density is higher. Therefore, we attribute the success of our correlation functional to its behavior in the intermediate region.
The tendency toward slowly varying electron gas, however, is not the only preferential behavior due to electron correlation. Because the strength of uniform electron gas correlation energy density 𝜀𝑐(𝑟𝑠) increases as a function of electron density; the system also prefers higher density. Since the total number of electrons is conserved via normalization, accumulating higher density in one region implies lower density in the adjacent areas, promoting higher gradient of electron density. Therefore, the tendency toward higher density may sometimes compete with the tendency toward homogeneity.
Most recently, it is shown [33] that for the large-Z limit of neutral atoms, electron correlation becomes local and slowly-varying. Further study is needed to address how the functional in this work performs in such limit, along with other important cases such as van der Waals interaction, hydrogen bonding, or bulk crystal.
4. Conclusions
Finally, we would like to emphasize that the simple behavior 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. 2(b), Fig. 3, and Fig. 4. 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 [34,10]. The true surprise nobody in the 90 year history of quantum mechanics would have suspected, however, is that the functional may be very simple both in the way it can be calculated and interpreted. We propose that the correlation energy is the energy that locally tends to drive the system toward the uniform electron gas.
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Supplementary Material See supplementary material for additional methods for spin polarized cases, computational details, and computer source codes.
Acknowledgements Support by Thailand Center of Excellence in Physics grant No. ThEP-60-PET-NU9 is gratefully acknowledged.
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Teepanis Chachiyo: Conceptualization, Methodology, Software, Writing- Original draft Hathaithip Chachiyo: Data curation, Validation, Writing- Reviewing and Editing,
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A very simple way to compute electron correlation energy Reasonably accurate for covalent bond molecules, competitive to B3LYP Hypothesis: the correlation energy favors uniform electron distribution