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Simple models for predicting correlation energy
Computational and Theoretical Chemistry 1067 (2015) 64–70
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Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
Simple models for predicting correlation energy Zuzana Is˘tvanková a, Victoria Grandy a, Raymond A. Poirier a,⇑, C. Dale Keefe b, Joshua W. Hollett c,d a
Department of Chemistry, Memorial University of Newfoundland, St. John’s, NL A1B 3X7, Canada Department of Chemistry, Cape Breton University, Sydney, NS B1P 6L2, Canada c Department of Chemistry, University of Winnipeg, Winnipeg, MB R3B 2G3, Canada d Department of Chemistry, University of Manitoba, Winnipeg, MB R3T 2N2, Canada b
a r t i c l e
i n f o
Article history: Received 25 March 2015 Received in revised form 20 April 2015 Accepted 20 April 2015 Available online 9 June 2015 Keywords: Electron correlation Hartree–Fock Inter-electronic distance Correlation energy models
a b s t r a c t A total of six different models that can account for the missing correlation energy in atomic systems were applied to the isoelectronic atomic series of two to ten electrons. Based on Rassolov’s linear correlation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi operator (Rassolov, 1999) we have defined a term, X ab ¼ Jab = T aa þ T bb , where J and T are the Coulomb repulsion and kinetic energy associated with orbitals a and b. This supports the idea that electron correlation depends on the separation of electrons in both position space and in momentum space. X ab , especially where a ¼ b, can account for the behaviour of electron correlation in these isoelectronic series. A model with only three terms (X 11 ; X 22 and J12 ) performed consistently well with the RMSD ranging from 9 104 ðN ¼ 3Þ to 6 105 ðN ¼ 5Þ. This study also provides insight into the Liu–Parr model which relates the correlation energy to the density at the nucleus (Liu and Parr, 2007). Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction
1.2. Models for the correlation energy in atomic systems
1.1. Electron correlation
For very small atomic systems it is possible to obtain near exact correlated wavefunctions. The first successful electronic structure computation of an explicitly correlated wavefunction was determined for the ground state of the helium atom by Hylleraas [4]. Hylleraas also expressed the exact energy of the helium-like ions as a Laurent series in nuclear charge Z using perturbation theory [5]. Following the Hylleraas ansatz, Davidson and co-workers estimated the expansion of the correlation energy for isoelectronic series of 2–18 electrons using experimental and ab initio data as [6,7]:
After almost a century since the postulation of quantum mechanics, except for fictitious two-electron systems [1,2], the only chemical systems that can be treated exactly are those with a single electron. For all many-electron problems we rely on approximate solutions to the electronic Schrödinger equation. The correlated motion of electrons remains one of the central unsolved challenges in quantum chemistry. In Hartree–Fock (HF) theory, the correlation of the electrons with opposite spin is completely neglected. The correlation energy is the difference between the exact non-relativistic energy, E, and the energy at the HF limit, EHFL .
Ecorr ¼ E EHFL
ð1Þ
Löwdin’s definition [3] of the correlation energy, based on a restricted HF wavefunction, refers to the missing Coulomb correlation only. The source of the Coulomb correlation arises from the electrons’ interaction through the repulsive electrostatic Coulomb potential, which is inversely proportional to the distance between the electrons, r 12 ¼ jr 1 r2 j.
Ecorr ðN; ZÞ ¼ A1 Z 1 þ A0 Z 0 þ A1 Z 1 þ A2 Z 2 þ
where the coefficients (Ai ), which depend on the number of electrons (N), are given by the difference between the coefficients in the expansion of the exact non-relativistic energy and the coefficients in the expansion of the HF energy [6]. Some of the earlier attempts at modelling the correlation energy in atomic systems explored the inverse relationship between the correlation energy and some measure of size [8–13]. Fröman [8] proposed a semi-empirical formula inversely relating correlation energy to the mean value of the HF inter-electronic distance hr 12 i and to a linear function of Z as:
Ecorr ¼ ⇑ Corresponding author. http://dx.doi.org/10.1016/j.comptc.2015.04.015 2210-271X/Ó 2015 Elsevier B.V. All rights reserved.
ð2Þ
1 1 ðaZ þ bÞ r 12
ð3Þ
Z. Is˘tvanková et al. / Computational and Theoretical Chemistry 1067 (2015) 64–70
Bernardi and co-workers [9] found that the correlation energy for helium-like atoms was proportional to the second moment of the electron density, hr2 i.
Table 2 The coefficientsa (in a.u.) in the correlation energy expansion in hri and the RMSD (Model 1). Series
Ecorr ¼
K ðZ rÞc hr 2 i
ð4Þ
in which r is a screening constant, K and c are fitted parameters. The dependence of the correlation energy on the size might arise naturally from the dependence of hr2 i on Z [9]. The correlation energy of neutral atoms in their ground state shows approximately linear dependence on the number of electrons, N, while a deviation from this linearity appears for positive and negative ions which increases with increasing charge [14]. Alonso and Cordero observed that the correlation energy and the number of antiparallel-spin electron pairs behave similarly with respect to Z, and they suggested that the correlation energy should be proportional to the number of opposite-spin electron pairs, N "# [15]. They extended the above to include the orbital dependency of pair energies, separating the contributions to the correlation energy into ss; sp, and pp terms (within the first two shells), restricted to electrons in the same shell. Koga et al. [16] pointed out that the electron density at the nucleus is almost the same for each cation, neutral atom and anion of the same atomic number. The electron density at the nucleus
Table 1 pffiffiffi The coefficientsa (in a.u.) in the correlation energy expansion in Z, J; T and the a RMSD (Model 1). N
2 3 4 5 6 7 8 9 10 All
Coefficients for expansion in Z A1
A0
0.0000 0.0000 0.0117 0.0069 0.0030 0.0001 0.0004 0.0007 0.0011 0.0000
0.0469 0.0540 0.0724 0.1296 0.1841 0.2337 0.2905 0.3393 0.3783 0.2501
b
A1 0.0112 0.0248 0.0949 0.1844 0.2624 0.3096 0.2848 0.1989 0.0096 0.6178
2 3 4 5 6 7 8 9 10 All
2 3 4 5 6 7 8 9 10 All a b c d
B0
0.0000 0.0000 0.0073 0.0029 0.0010 0.0001 0.0001 0.0001 0.0001 0.0013
0.0466 0.0536 0.0813 0.1319 0.1782 0.2204 0.2788 0.3327 0.3818 0.1867
C1
C0
0.0000 0.0000 0.0105 0.0059 0.0027 0.0001 0.0001 0.0001 0.0002 0.0008
0.0467 0.0529 0.0775 0.1311 0.1851 0.2311 0.2995 0.3611 0.4148 0.1938
B1
0.0059 0.0249 0.0931 0.2418 0.4613 0.6988 0.7528 0.6121 0.0001 0.9724 pffiffiffi Coefficients for expansion in T d C 1 0.0097 0.0202 0.0830 0.1651 0.2411 0.2960 0.3723 0.3931 0.3386 0.1212
RMSD is the root mean square deviation. Eq. (2). Eq. (8). Eq. (9).
2e 3e 4e 5e 6e 7e 8e 9e 10e a
Coefficients
RMSD
D2
D1
D0
D1
0.0004 0.0002 0.0002 0.0003 0.0003 0.0003 0.0006 0.0008 0.0008
0.0032 0.0025 0.0050 0.0068 0.0077 0.0082 0.0110 0.0119 0.0109
0.0467 0.0534 0.0864 0.1375 0.1850 0.2298 0.3036 0.3690 0.4251
0.0000 0.0000 0.1762 0.1394 0.0798 0.0048 0.0011 0.0009 0.0022
3.1E06 1.1E05 3.1E05 3.3E05 5.5E05 8.5E05 7.1E05 2.8E05 9.8E05
Eq. (10).
can be modelled almost entirely in terms of the s-type atomic orbitals, predominantly the 1s orbital. They showed that for the neutral atoms, and singly charged anions and cations, the hydrogenic approximation of each 1s electron contributing Z 3 =p to the electron density at the nucleus reproduces HF values within 11% accuracy [16]. Following these results the idea that the correlation energy is related to the electrostatic potential at the nucleus was introduced by Liu and Parr [17]. They report a simple empirical formula relating the electron density at the nucleus, qð0Þ, and the total electron correlation energy of atoms.
Ecorr ¼ C LP
Nqð0Þ Z cLP
ð5Þ
RMSD A2 0.0033 0.0075 0.0181 0.0482 0.0720 0.0844 0.0534 0.0325 0.0082
3.6E10 6.6E10 1.6E09 2.0E08 1.7E08 2.2E09 2.8E07 7.6E07 1.2E06 1.7E02
Coefficients for expansion in Jc B1
65
C LP and cLP are parameters optimized for each isoelectronic atomic series, where both C LP and cLP are irregular functions of N. The accuracy of the simple relationship obtained in Liu and Parr’s work is quite encouraging, especially if there is a way to extend these simple models to calculations on larger chemical systems. McCarthy and Thakkar [18] proposed a model for the correlation energy of neutral atoms and singly charged cations as:
Ecorr
! Nintra Nintra 1 N "" 1 "# "# 3=2 ¼a N þb 1 þc N"# N"# hriHF N "# hriHF
ð6Þ
B2 0.0012 0.0136 0.0077 0.0343 0.0589 0.0736 0.0784 0.0691 0.0013 0.9423
6.6E12 2.2E10 6.1E08 1.1E07 1.3E07 2.3E07 6.8E07 1.0E06 1.2E06 7.4E03
C 2 0.0032 0.0023 0.0269 0.0544 0.1678 0.4095 0.8132 1.0371 1.0536 0.8745
2.0E12 4.3E09 9.8E09 2.5E09 1.6E09 2.9E09 3.2E09 1.9E09 4.6E09 1.8E02
The contribution to the total correlation energy from the antiparallel spin electron pairs depends on
Nintra "# , N "#
the ratio of the number of
intra-shell antiparallel spin electron pairs to the number of the total antiparallel spin electron pairs, while the contribution from the parN
, in which N "" is the allel spin electron pairs depends on the ratio N"" "# total number of same spin electron pairs. hriHF is the average value of r, which contains the model’s dependence on Z [18]. Rassolov [19] proposed a linear correlation operator which modifies the HF Hamiltonian and gives the correlation energy for a single determinantal wavefunction: 0 b ¼ CR C 2 r 12 p212
ð7Þ
where r 12 is the distance between the electrons, p12 is the relative momentum and C 0R is a numerical constant. The operator has a larger expectation value when electrons approach each other (r 212 ! 0), and for electrons that move along with each other (p212 ! 0), than for the electrons with large separation and high relative momentum. Electrons that move fast with respect to each other spend less time in each other’s vicinity [19].
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Table 3 The parameters for the Liu–Parr model and Model 2 along with the RMSD (in a.u.). N
2 3 4 5 6 7 8 9 10 a b c
Liu–Parr modela
Liu–Parr modelb
Model 2c
C LP
cLP
C LP
cLP
RMSD
C2
c2
RMSD
0.0467 0.0302 0.0166 0.0209 0.0292 0.0446 0.0518 0.0591 0.0664
3.0796 3.0381 2.3632 2.5655 2.7720 2.9838 2.9907 3.0079 3.0285
0.04719 0.03113 0.01512 0.02079 0.03034 0.04536 0.05430 0.06198 0.07069
3.06966 3.03265 2.32397 2.55736 2.77901 2.98339 2.99997 3.01728 3.04289
1.0E03 3.3E04 5.1E03 4.5E03 2.2E03 4.3E04 6.7E04 1.4E03 2.3E03
0.0432 0.0451 0.0289 0.0439 0.0776 0.1450 0.2009 0.2600 0.3291
0.0273 0.0500 0.6893 0.4682 0.2669 0.0938 0.0746 0.0580 0.0398
3.3E04 5.2E04 5.0E03 4.0E03 1.5E03 1.5E03 7.9E04 6.6E04 1.2E03
Liu–Parr model, Eq. (5) [17]. Liu–Parr model, Eq. (5), using the pVTZ basis set. Model 2, Eq. (13).
Fig. 1. The change in the density at the nucleus, Dqð0ÞN (Eq. (14)), for the N ¼ 3 and 4 (a) and, the N = 5–10 (b) series as a function of Z.
Fig. 2. The change in DJsN (Eq. (15)), for the N ¼ 3 and 4 (a) and, the N = 5–10 (b) series as a function of Z.
1.3. Modelling of the correlation energy in atomic systems 2. Computational method The focus of this study is to report empirical models that are able to accurately predict the electron correlation energy of atomic systems. We also show how new models can be derived from previously reported models. Our models are based on properties (e.g., Coulomb, exchange and kinetic energies) that can be easily calculated using HF theory and that would also be applicable to molecules. These correlation models can aid in the development of new theoretical methods that can treat electron correlation more accurately and efficiently. At the same time, these simple models inevitably add to the understanding of electron correlation in atoms. It is interesting to note that these very simple, few-parameter, atomic models have an accuracy comparable to some of the more complex and computationally more expensive post-HF methods. The simple electron correlation models may prove quite useful in the study of large chemical systems by offering higher accuracy compared to existing empirical methods.
All the calculations were performed with MUNgauss [20]. The atomic energies were calculated at the HF level of theory (RHF for closed shell and ROHF for open shell systems) using a basis set which we will refer to as pVTZ. The basis set is based on the aug-cc-pVTZ basis set as implemented in Gaussian03 [21], where the scale factors were optimized for each system. For a given N, the scale factors were found to converge to a linear function of Z. For that reason the scale factors were constrained to be a linear function of Z. The basis set, the equations used to define the scale factors for the systems studied, along with all the data used in this study are given in the supporting information. The basis set used in the calculations is large enough that the difference between any calculated HF energy and the exact non-relativistic energy will be mostly due to correlation energy, and thus the basis set incompleteness error will be negligible for the systems studied. The correlation energies for the atomic systems are taken from the
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Fig. 3. The change in correlation energy, DENcorr ¼ ENcorr EN1 corr , for the N ¼ 3 and 4 (a) and, the N = 5–10 (b) series as a function of Z.
summary given by Chakravorty et al. [6,7] Throughout this paper the subscripts 1, 2, 3, 4 and 5 refer to the 1s; 2s; 2px ; 2py and 2pz orbitals, respectively. 3. Modelling of the atomic systems 3.1. Expansion of the electron correlation as a function of J;
pffiffiffi T and hri
The behaviour of the correlation energy as a function of N and Z is illustrated in Figs. S4 and S5, respectively, for the 2–18 electron series (see supporting information). The total Coulomb energy (J), exchange energy (K), potential energy (V), square root of the pffiffiffi kinetic energy ( T ), as well as all energy terms pffiffiffiffiffiffiffi ðJ ab ; K ab ; V ab ; T ab Þ, are all linear functions of Z (Figs. S1–S3). The inverse of the average value of r (hri1 ) is also a linear function of pffiffiffi Z (Fig. S10). Using these relationships, the substitution of J; T and hri for Z in the correlation energy expansion in Z (based on Eq. (2)) [6] results in: Model 1
Fig. 4. X 11 as a function of Z (a) and the relationship between Ecorr and X 11 (b) for the N ¼ 2 series.
coefficients resulted in a RMSD of 102 . Similar expansions in terms of hri are given in Table 2. In that case the N = 4–6 isoelectronic series have a significant contribution from the hri1 term. It is interesting to note that, for systems with low charge, the correlation energy P correlates well with a J aa where, J aa is the Coulomb repulsion between electrons in orbital a and the sum is over all doubly occupied orbitals. Deviations start occurring for ions with charges greater than five (Fig. S9). 3.2. Extending the Liu–Parr model The linear dependence of J on Z could also be used to modify the correlation energy model described by Liu and Parr [17]. For the H-atom and one-electron ions, the density at the nucleus, qð0Þ, is related to Z as:
qð0Þ / Z 3
ð11Þ 2
Furthermore, for all systems studied, we found Nqð0Þ=Z to be linearly related to J. Based on these relationships the Liu–Parr model, Eq. (5), can be expressed as, 3cLP
Ecorr ðN; JÞ ¼ B1 J 1 þ B0 J 0 þ B1 J 1 þ B2 J2 pffiffiffi pffiffiffi1 pffiffiffi0 pffiffiffi1 pffiffiffi2 Ecorr ðN; T Þ ¼ C 1 T þ C 0 T þ C 1 T þ C 2 T Ecorr ðN; hriÞ ¼ D2 hri2 þ D1 hri1 þ D0 hri0 þ D1 hri1
ð8Þ
Ecorr ¼ C LP ðaJ J þ bJ Þ
ð9Þ ð10Þ
pffiffiffi One advantage of the expansion of the correlation energy in J; T or hri over the expansion in Z is that it could also be applicable to modelling correlation energy of molecular systems. The optimized coefficients for the truncated correlation energy expansion in Z are summarized and compared to the coefficients in the correlation pffiffiffi energy expansions in J and T in Table 1. As expected, the coefficients Ai ; Bi and C i ði ¼ 1; 2; 3; 4Þ behave similarly with a change in N. For all three cases there is a much larger contribution from the pffiffiffi first order term ðZ; J; T Þ for the N ¼ 4; 5 and 6 isoelectronic series. Attempts to fit the N = 2–10 electron series with a single set of
67
Fig. 5. X 11 as a function of Z for the N = 2–10 series.
ð12Þ
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We found that Eq. (12) can be simplified without affecting the quality of the fits to, Model 2
Ecorr ¼ C 2 J c2
ð13Þ
As seen in Table 3 the parameters obtained using the pVTZ basis set are similar to those obtained by Liu and Parr [17]. In both the Liu– Parr model and Model 2 the parameters depend on N. Similar trends in the behaviour of the c2 and cLP parameters (Figs. S6 and S7) and the C 2 and C LP parameters (Fig. S8) with the increasing number of electrons suggest that these models are inter-related due to the relationship between Z and J. Model 2, a function of the total Coulomb energy only, performs as well as the Parr–Liu model. The relationship between the coefficients in the above models and N follows a similar trend. Although, the coefficients Ai ; Bi pffiffiffi and C i in the correlation energy expansion in Z; J and T , respectively, and the parameters C j and cj in Model 2 and in the Liu–Parr model could be approximated for all N, the extrapolation implemented this way would not lead to accurate results. For each model the parameters exhibit irregular behaviour with respect to N. The correlation energy is not expected to be a smooth function of N, although, it likely behaves smoothly in certain N intervals, such as for N = 5–7, N = 7–10, or for N = 13–15 and N = 15–18, for example (Fig. S4).
series) and very little change when electrons are paired. This would suggest that for Model 2 the total Coulomb repulsion in Eq. (13) could be replaced by JsN . As seen in the supporting information such a model performs equally well. 3.3. Correlation energy as a function of Coulomb and kinetic energies Rassolov [19] proposed a linear correlation operator which approximates the correlation energy for the single determinantal wavefunction based on its dependence on the separation of two electrons and their relative momentum (Eq. (7)). Starting with Rassolov’s operator, it is possible to devise other simple electron correlation models based on the properties of HF orbitals and the total wavefunction. Replacing the expectation values of the inter-electronic separation r212 and the relative momentum p212 operators in Rassolov’s correlation operator by the average inter-particle distance [12] and by the average inter-particle momentum, respectively, we arrive at:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi habjr 212 jabi dr ab qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ habjp212 jabi dpab
ðdr12 Þab ¼
ð16Þ
ðdp12 Þab
ð17Þ
and
Eab corr ¼ 3.2.1. The behaviour of qð0Þ and Coulomb repulsion As shown in Fig. 1 it is interesting to note that N
N
Dqð0Þ ¼ qð0Þ qð0Þ
N1
ð14Þ
increases for N ¼ 3 and 4 (Fig. 1(a)) but decreases for N = 6–10 (Fig. 1(b)). For N ¼ 5; Dqð0Þ5 initially decreases and starts increasing at higher values of Z. There is a clear distinction when adding electrons to s orbitals, which causes an increase in qð0Þ and adding electrons to p orbitals, which causes a decrease in qð0Þ. Because of its relationship with the density at the nucleus, a similar behaviour is expected for the change in Coulomb repulsion. The change in the total Coulomb repulsion associated with only the 1s and 2s electrons is defined as,
DJsN ¼ JsN JsN1
ð15Þ
As seen in Fig. 2(a) change in density at the nucleus does indeed translate to a corresponding change in JsN . In that case DJsN increases linearly for N ¼ 3 and N ¼ 4 (Fig. 2(a)) and decreases for N = 5 to N = 10 (Fig. 2(b)). The equivalent plots for the change in Ecorr are shown in Fig. 3. Fig. 3(a) clearly shows the large (and linear) increase in electron correlation when filling the 2s orbital (N ¼ 4 series). Adding electrons to the 2p orbitals (Fig. 3(b)), on the other hand, causes a decrease in electron correlation (N = 5, 6 and 7
C ab ðdr 12 Þab ðdp12 Þab
ð18Þ
Poirier and Hollett [12] observed that the average distance between two electrons, one in occupied molecular orbital a and one in occupied molecular orbital b; ðdr12 Þab is inversely proportional to the two-electron Coulomb integral,
ðdr12 Þab ¼
a
ð19Þ
J ab
Using the above relationship for the average inter-particle distance and a similar expression for the average inter-particle momentum, with me ¼ 1 a:u:,
ðdp12 Þab ¼
pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T aa þ T bb
ð20Þ
The correlation energy for a given electron pair associated with molecular orbitals a and b becomes:
J ab Eab corr ¼ C ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T aa þ T bb
ð21Þ
The total correlation energy can be written as a sum of the pair correlation energies:
Ecorr ¼
X J ab C ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T aa þ T bb ab
ð22Þ
Table 4 The optimized coefficients and the RMSD (in a.u.) for Models 3, 4 and 5. N
2 3 4 5 6 7 8 9 10 a b c
Eq. (25). Eq. (26). Eq. (27).
Model 3a
Model 4b
Model 5c
aX11 3
RMSD
aX11 4
aJ11 4
RMSD
aX11 5
aX22 5
aJ12 5
RMSD
0.0734 0.0825 0.4049 0.3669 0.3483 0.3435 0.4487 0.5474 0.6423
7.3E04 1.6E03 8.9E02 5.5E02 2.9E02 9.7E03 9.3E03 8.2E03 6.2E03
0.0734 0.0781 0.0945 0.1615 0.2318 0.3041 0.4069 0.5070 0.6090
0.0001 0.0003 0.0198 0.0127 0.0070 0.0023 0.0024 0.0022 0.0018
4.4E04 8.9E04 2.7E03 3.8E03 4.0E03 3.5E03 2.6E03 1.7E03 9.3E04
0.0734 0.0784 0.3299 0.6903 0.9711 1.1701 1.2783 1.3996 0.8936
– – 0.4002 0.9774 1.3993 1.6624 1.6765 1.7242 0.5439
– 0.0004 0.0141 0.0083 0.0037 0.0001 0.0003 0.0004 0.0009
7.3E04 8.8E04 2.8E04 6.1E05 1.4E04 8.4E05 2.5E04 2.3E04 8.0E04
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Table 5 The optimized parameters and RMSD (in a.u.) for Model 6.a N
2 3 4 5 6 7 8 9 10 a b c
Coefficients
RMSD
aX11 6
aX12 6
aX22 6
aX13 6
aX23 6
b aX34 6
0.0734 0.0549 0.3180 0.5499 1.0589 2.1462 2.6022 3.2960 3.4503
– 0.0341 0.1463 0.0792 0.1900 0.7531 0.8195 0.9523 0.8497
– – 0.4990 1.1432 0.8581 0.2644 0.0 0.0064 0.0
– – – 0.0861 0.2535 0.4622 0.2725 0.2355 0.1551
– – – 0.1816 0.3076 0.0872 0.4181 0.4980 0.4997
– – – – 2.0651 1.1941 0.0294 0.0614 0.1329
c aX33 6
aK6 – –
– 0.0 0.0 0.2321 0.0052 0.0088
– 0.0018 0.2679 0.1595 0.0718 0.0003 0.0012 0.0013 0.0003
7.3E04 1.2E04 1.7E05 1.9E05 6.9E06 3.2E05 7.5E05 5.2E05 2.7E05
Eq. (28). X 34 ¼ X 35 ¼ X 45 . X 33 ¼ X 44 ¼ X 55 .
where C ab would differ from the value of C 0R in the Rassolov operator, and the parameter C ab depends on the number of electrons in the system. For simplicity, a new variable X ab is defined as,
J ab X ab ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T aa þ T bb
ð23Þ
and the relationship between the correlation energy and the variable X ab is examined. For the two-electron series,
Ecorr / X 11
ð24Þ
In this case, the behaviour of X 11 with increasing Z (Fig. 4(a)) resembles that of the correlation energy with the increasing Z (Fig. S5). The relation between Ecorr and X 11 is also shown in Fig. 4(b). Interestingly, X 11 for the other isoelectronic series (Fig. 5) also resembles the behaviour of the correlation energy with increasing nuclear charge, except for the atomic series containing four, five, and six electrons. It was therefore initially investigated whether the total correlation energy for the atomic systems could be approximated using only X 11 . Model 3
Ecorr ¼ aX11 3 X 11
ð25Þ
Based on the Liu–Parr model, which suggests that most of the electron correlation can be accounted for by the core electrons, we tested two additional simple models. In these models only terms involving 1s and 2s orbitals are included and the intercept is taken to be zero. Model 4 J11 Ecorr ¼ aX11 4 X 11 þ a4 J 11
ð26Þ
Model 5 J12 X22 Ecorr ¼ aX11 5 X 11 þ a5 X 22 þ a5 J 12
ð27Þ
The resulting fits and the optimized parameters are summarized in Table 4. Model 3 does very poorly at describing the N = 4, 5 and 6 electron series. As expected, based on Table 1 and Eq. (8), adding the J11 term reduces the RMSD by an order of magnitude for the N = 4, 5 and 6 electron series. The RMSD is further reduced for Model 5. This simple model with only three parameters performs well for all the isoelectronic series. This is consistent with the trends observed in the change in electron correlation (Fig. 3) and with the change in the density at the nucleus, Fig. 1. For all cases Model 5 recovers on average 100% of the correlation energy. The correlation energy is overestimated (107%) or underestimated (98%) only for some of the N ¼ 2 and N ¼ 3 ions. To determine whether there is any advantage to be gained by increasing the number of empirical parameters we investigated the following model,
Model 6
Ecorr ¼
X K aXab 6 X ab þ a6 K 12
ð28Þ
ab
The correlation energy is expressed as a sum of the pair correlation energies, where each electron pair is characterized by a specific value of X ab and a K 12 term. For Model 6 all coefficients are allowed to vary independently. The results are summarized in Table 5. Although Model 6 is successful in predicting accurate correlation energies with lower RMSDs it provides very little new insight. 4. Conclusions A total of six different models were investigated for the N = 2– 10 isoelectronic series. The performance of the correlation energy models for the atomic systems was assessed using the RMSD. Many of the models presented here can successfully account for the missing correlation energy in the atomic isoelectronic series. Analysis of DENcorr ; DqN ð0Þ and DJ sN shows that there is a clear distinction between filling the 2s orbital and the 2p orbitals. Filling the 2s orbital causes an increase in electron correlation while adding unpaired electrons to the 2p orbitals (N = 5, 6, 7) causes a decrease in electron correlation. On the other hand, when further electrons are added to the 2p orbitals the change in correlation energy is small (N = 8, 9, 10). This provides a new perspective to the different behaviour of the N = 4 to N = 6 electron series in all the models. Model 5, with only three terms (X 11 ; X 22 and J 12 ) performed consistently well with the RMSD ranging from 9 104 (N ¼ 3) to 6 105 (N ¼ 5). Consistent with Rassolov’s linear corpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relation operator [19], X ab ¼ J ab = T aa þ T bb , especially where, a ¼ b, can account for the behaviour of electron correlation in these isoelectronic series. The advantages of the correlation models presented here is their computational efficiency and possible applicability to a wide range of larger chemical system. Acknowledgements We gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Memorial University of Newfoundland, Compute Canada and the Atlantic Excellence Network (ACEnet), regional high performance computing consortium for universities in Atlantic Canada. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.comptc.2015.04. 015.
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[13] M.A. Robb, W.J. Haines, I.G. Csizmadia, Theoretical definition of size of electron pairs and its stereochemical implications, J. Am. Chem. Soc. 95 (1) (1973) 42–48. [14] J. Alonso, N. Cordero, Variation of the ground-state correlation energy of atoms with atomic number, J. Phys. B: At. Mol. Opt. Phys. 29 (9) (1996) 1629–1636. [15] J. Alonso, N.H. March, N. Cordero, A. Rubio, Correlation energies of light atoms related to pairing between antiparallel spin electrons, J. Phys. B: At. Mol. Opt. Phys. 36 (13) (2003) 2695–2705. [16] T. Koga, S. Watanabe, K. Kanayama, R. Yasuda, A.J. Thakkar, Improved Roothaan–Hartree–Fock wave-functions for atoms and ions with n-lessthan-or-equal-to-54, J. Chem. Phys. 103 (8) (1995) 3000–3005. [17] S. Liu, R.G. Parr, Atomic correlation energy from the electron density at the nucleus, J. Phys. Chem. A 111 (2007) 10422–10425. [18] S.P. McCarthy, A.J. Thakkar, Simple models for electron correlation energies in atoms, Chem. Phys. Lett. 494 (2010) 312–314. [19] V.A. Rassolov, An ab initio linear electron correlation functional, J. Chem. Phys. 110 (8) (1999) 3672–3677. [20] R.A. Poirier, J.W. Hollett, P.L. Warburton, MUNgauss, Memorial University, Chemistry Department, St. John’s, NL A1B 3X7. With contributions from A. Alrawashdeh, J.-P. Becker, J. Besaw, S.D. Bungay, F. Colonna, A. El-Sherbiny, T. Gosse, D. Keefe, A. Kelly, D. Nippard, C.C. Pye, D. Reid, K. Saputantri, M. Shaw, M. Staveley, O. Stueker, Y. Wang, J. Xidos. [21] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery, J.T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, X.L.H. Nakai, M. Klene, J.E. Knox, H.P. Hratchian, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, S.D.V.G. Zakrzewski, M.C.S.A.D. Daniels, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, D.J.F.R.L. Martin, T. Keith, M.A. AlLaham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Go.
Contents lists available at ScienceDirect
Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
Simple models for predicting correlation energy Zuzana Is˘tvanková a, Victoria Grandy a, Raymond A. Poirier a,⇑, C. Dale Keefe b, Joshua W. Hollett c,d a
Department of Chemistry, Memorial University of Newfoundland, St. John’s, NL A1B 3X7, Canada Department of Chemistry, Cape Breton University, Sydney, NS B1P 6L2, Canada c Department of Chemistry, University of Winnipeg, Winnipeg, MB R3B 2G3, Canada d Department of Chemistry, University of Manitoba, Winnipeg, MB R3T 2N2, Canada b
a r t i c l e
i n f o
Article history: Received 25 March 2015 Received in revised form 20 April 2015 Accepted 20 April 2015 Available online 9 June 2015 Keywords: Electron correlation Hartree–Fock Inter-electronic distance Correlation energy models
a b s t r a c t A total of six different models that can account for the missing correlation energy in atomic systems were applied to the isoelectronic atomic series of two to ten electrons. Based on Rassolov’s linear correlation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi operator (Rassolov, 1999) we have defined a term, X ab ¼ Jab = T aa þ T bb , where J and T are the Coulomb repulsion and kinetic energy associated with orbitals a and b. This supports the idea that electron correlation depends on the separation of electrons in both position space and in momentum space. X ab , especially where a ¼ b, can account for the behaviour of electron correlation in these isoelectronic series. A model with only three terms (X 11 ; X 22 and J12 ) performed consistently well with the RMSD ranging from 9 104 ðN ¼ 3Þ to 6 105 ðN ¼ 5Þ. This study also provides insight into the Liu–Parr model which relates the correlation energy to the density at the nucleus (Liu and Parr, 2007). Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction
1.2. Models for the correlation energy in atomic systems
1.1. Electron correlation
For very small atomic systems it is possible to obtain near exact correlated wavefunctions. The first successful electronic structure computation of an explicitly correlated wavefunction was determined for the ground state of the helium atom by Hylleraas [4]. Hylleraas also expressed the exact energy of the helium-like ions as a Laurent series in nuclear charge Z using perturbation theory [5]. Following the Hylleraas ansatz, Davidson and co-workers estimated the expansion of the correlation energy for isoelectronic series of 2–18 electrons using experimental and ab initio data as [6,7]:
After almost a century since the postulation of quantum mechanics, except for fictitious two-electron systems [1,2], the only chemical systems that can be treated exactly are those with a single electron. For all many-electron problems we rely on approximate solutions to the electronic Schrödinger equation. The correlated motion of electrons remains one of the central unsolved challenges in quantum chemistry. In Hartree–Fock (HF) theory, the correlation of the electrons with opposite spin is completely neglected. The correlation energy is the difference between the exact non-relativistic energy, E, and the energy at the HF limit, EHFL .
Ecorr ¼ E EHFL
ð1Þ
Löwdin’s definition [3] of the correlation energy, based on a restricted HF wavefunction, refers to the missing Coulomb correlation only. The source of the Coulomb correlation arises from the electrons’ interaction through the repulsive electrostatic Coulomb potential, which is inversely proportional to the distance between the electrons, r 12 ¼ jr 1 r2 j.
Ecorr ðN; ZÞ ¼ A1 Z 1 þ A0 Z 0 þ A1 Z 1 þ A2 Z 2 þ
where the coefficients (Ai ), which depend on the number of electrons (N), are given by the difference between the coefficients in the expansion of the exact non-relativistic energy and the coefficients in the expansion of the HF energy [6]. Some of the earlier attempts at modelling the correlation energy in atomic systems explored the inverse relationship between the correlation energy and some measure of size [8–13]. Fröman [8] proposed a semi-empirical formula inversely relating correlation energy to the mean value of the HF inter-electronic distance hr 12 i and to a linear function of Z as:
Ecorr ¼ ⇑ Corresponding author. http://dx.doi.org/10.1016/j.comptc.2015.04.015 2210-271X/Ó 2015 Elsevier B.V. All rights reserved.
ð2Þ
1 1 ðaZ þ bÞ r 12
ð3Þ
Z. Is˘tvanková et al. / Computational and Theoretical Chemistry 1067 (2015) 64–70
Bernardi and co-workers [9] found that the correlation energy for helium-like atoms was proportional to the second moment of the electron density, hr2 i.
Table 2 The coefficientsa (in a.u.) in the correlation energy expansion in hri and the RMSD (Model 1). Series
Ecorr ¼
K ðZ rÞc hr 2 i
ð4Þ
in which r is a screening constant, K and c are fitted parameters. The dependence of the correlation energy on the size might arise naturally from the dependence of hr2 i on Z [9]. The correlation energy of neutral atoms in their ground state shows approximately linear dependence on the number of electrons, N, while a deviation from this linearity appears for positive and negative ions which increases with increasing charge [14]. Alonso and Cordero observed that the correlation energy and the number of antiparallel-spin electron pairs behave similarly with respect to Z, and they suggested that the correlation energy should be proportional to the number of opposite-spin electron pairs, N "# [15]. They extended the above to include the orbital dependency of pair energies, separating the contributions to the correlation energy into ss; sp, and pp terms (within the first two shells), restricted to electrons in the same shell. Koga et al. [16] pointed out that the electron density at the nucleus is almost the same for each cation, neutral atom and anion of the same atomic number. The electron density at the nucleus
Table 1 pffiffiffi The coefficientsa (in a.u.) in the correlation energy expansion in Z, J; T and the a RMSD (Model 1). N
2 3 4 5 6 7 8 9 10 All
Coefficients for expansion in Z A1
A0
0.0000 0.0000 0.0117 0.0069 0.0030 0.0001 0.0004 0.0007 0.0011 0.0000
0.0469 0.0540 0.0724 0.1296 0.1841 0.2337 0.2905 0.3393 0.3783 0.2501
b
A1 0.0112 0.0248 0.0949 0.1844 0.2624 0.3096 0.2848 0.1989 0.0096 0.6178
2 3 4 5 6 7 8 9 10 All
2 3 4 5 6 7 8 9 10 All a b c d
B0
0.0000 0.0000 0.0073 0.0029 0.0010 0.0001 0.0001 0.0001 0.0001 0.0013
0.0466 0.0536 0.0813 0.1319 0.1782 0.2204 0.2788 0.3327 0.3818 0.1867
C1
C0
0.0000 0.0000 0.0105 0.0059 0.0027 0.0001 0.0001 0.0001 0.0002 0.0008
0.0467 0.0529 0.0775 0.1311 0.1851 0.2311 0.2995 0.3611 0.4148 0.1938
B1
0.0059 0.0249 0.0931 0.2418 0.4613 0.6988 0.7528 0.6121 0.0001 0.9724 pffiffiffi Coefficients for expansion in T d C 1 0.0097 0.0202 0.0830 0.1651 0.2411 0.2960 0.3723 0.3931 0.3386 0.1212
RMSD is the root mean square deviation. Eq. (2). Eq. (8). Eq. (9).
2e 3e 4e 5e 6e 7e 8e 9e 10e a
Coefficients
RMSD
D2
D1
D0
D1
0.0004 0.0002 0.0002 0.0003 0.0003 0.0003 0.0006 0.0008 0.0008
0.0032 0.0025 0.0050 0.0068 0.0077 0.0082 0.0110 0.0119 0.0109
0.0467 0.0534 0.0864 0.1375 0.1850 0.2298 0.3036 0.3690 0.4251
0.0000 0.0000 0.1762 0.1394 0.0798 0.0048 0.0011 0.0009 0.0022
3.1E06 1.1E05 3.1E05 3.3E05 5.5E05 8.5E05 7.1E05 2.8E05 9.8E05
Eq. (10).
can be modelled almost entirely in terms of the s-type atomic orbitals, predominantly the 1s orbital. They showed that for the neutral atoms, and singly charged anions and cations, the hydrogenic approximation of each 1s electron contributing Z 3 =p to the electron density at the nucleus reproduces HF values within 11% accuracy [16]. Following these results the idea that the correlation energy is related to the electrostatic potential at the nucleus was introduced by Liu and Parr [17]. They report a simple empirical formula relating the electron density at the nucleus, qð0Þ, and the total electron correlation energy of atoms.
Ecorr ¼ C LP
Nqð0Þ Z cLP
ð5Þ
RMSD A2 0.0033 0.0075 0.0181 0.0482 0.0720 0.0844 0.0534 0.0325 0.0082
3.6E10 6.6E10 1.6E09 2.0E08 1.7E08 2.2E09 2.8E07 7.6E07 1.2E06 1.7E02
Coefficients for expansion in Jc B1
65
C LP and cLP are parameters optimized for each isoelectronic atomic series, where both C LP and cLP are irregular functions of N. The accuracy of the simple relationship obtained in Liu and Parr’s work is quite encouraging, especially if there is a way to extend these simple models to calculations on larger chemical systems. McCarthy and Thakkar [18] proposed a model for the correlation energy of neutral atoms and singly charged cations as:
Ecorr
! Nintra Nintra 1 N "" 1 "# "# 3=2 ¼a N þb 1 þc N"# N"# hriHF N "# hriHF
ð6Þ
B2 0.0012 0.0136 0.0077 0.0343 0.0589 0.0736 0.0784 0.0691 0.0013 0.9423
6.6E12 2.2E10 6.1E08 1.1E07 1.3E07 2.3E07 6.8E07 1.0E06 1.2E06 7.4E03
C 2 0.0032 0.0023 0.0269 0.0544 0.1678 0.4095 0.8132 1.0371 1.0536 0.8745
2.0E12 4.3E09 9.8E09 2.5E09 1.6E09 2.9E09 3.2E09 1.9E09 4.6E09 1.8E02
The contribution to the total correlation energy from the antiparallel spin electron pairs depends on
Nintra "# , N "#
the ratio of the number of
intra-shell antiparallel spin electron pairs to the number of the total antiparallel spin electron pairs, while the contribution from the parN
, in which N "" is the allel spin electron pairs depends on the ratio N"" "# total number of same spin electron pairs. hriHF is the average value of r, which contains the model’s dependence on Z [18]. Rassolov [19] proposed a linear correlation operator which modifies the HF Hamiltonian and gives the correlation energy for a single determinantal wavefunction: 0 b ¼ CR C 2 r 12 p212
ð7Þ
where r 12 is the distance between the electrons, p12 is the relative momentum and C 0R is a numerical constant. The operator has a larger expectation value when electrons approach each other (r 212 ! 0), and for electrons that move along with each other (p212 ! 0), than for the electrons with large separation and high relative momentum. Electrons that move fast with respect to each other spend less time in each other’s vicinity [19].
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66
Table 3 The parameters for the Liu–Parr model and Model 2 along with the RMSD (in a.u.). N
2 3 4 5 6 7 8 9 10 a b c
Liu–Parr modela
Liu–Parr modelb
Model 2c
C LP
cLP
C LP
cLP
RMSD
C2
c2
RMSD
0.0467 0.0302 0.0166 0.0209 0.0292 0.0446 0.0518 0.0591 0.0664
3.0796 3.0381 2.3632 2.5655 2.7720 2.9838 2.9907 3.0079 3.0285
0.04719 0.03113 0.01512 0.02079 0.03034 0.04536 0.05430 0.06198 0.07069
3.06966 3.03265 2.32397 2.55736 2.77901 2.98339 2.99997 3.01728 3.04289
1.0E03 3.3E04 5.1E03 4.5E03 2.2E03 4.3E04 6.7E04 1.4E03 2.3E03
0.0432 0.0451 0.0289 0.0439 0.0776 0.1450 0.2009 0.2600 0.3291
0.0273 0.0500 0.6893 0.4682 0.2669 0.0938 0.0746 0.0580 0.0398
3.3E04 5.2E04 5.0E03 4.0E03 1.5E03 1.5E03 7.9E04 6.6E04 1.2E03
Liu–Parr model, Eq. (5) [17]. Liu–Parr model, Eq. (5), using the pVTZ basis set. Model 2, Eq. (13).
Fig. 1. The change in the density at the nucleus, Dqð0ÞN (Eq. (14)), for the N ¼ 3 and 4 (a) and, the N = 5–10 (b) series as a function of Z.
Fig. 2. The change in DJsN (Eq. (15)), for the N ¼ 3 and 4 (a) and, the N = 5–10 (b) series as a function of Z.
1.3. Modelling of the correlation energy in atomic systems 2. Computational method The focus of this study is to report empirical models that are able to accurately predict the electron correlation energy of atomic systems. We also show how new models can be derived from previously reported models. Our models are based on properties (e.g., Coulomb, exchange and kinetic energies) that can be easily calculated using HF theory and that would also be applicable to molecules. These correlation models can aid in the development of new theoretical methods that can treat electron correlation more accurately and efficiently. At the same time, these simple models inevitably add to the understanding of electron correlation in atoms. It is interesting to note that these very simple, few-parameter, atomic models have an accuracy comparable to some of the more complex and computationally more expensive post-HF methods. The simple electron correlation models may prove quite useful in the study of large chemical systems by offering higher accuracy compared to existing empirical methods.
All the calculations were performed with MUNgauss [20]. The atomic energies were calculated at the HF level of theory (RHF for closed shell and ROHF for open shell systems) using a basis set which we will refer to as pVTZ. The basis set is based on the aug-cc-pVTZ basis set as implemented in Gaussian03 [21], where the scale factors were optimized for each system. For a given N, the scale factors were found to converge to a linear function of Z. For that reason the scale factors were constrained to be a linear function of Z. The basis set, the equations used to define the scale factors for the systems studied, along with all the data used in this study are given in the supporting information. The basis set used in the calculations is large enough that the difference between any calculated HF energy and the exact non-relativistic energy will be mostly due to correlation energy, and thus the basis set incompleteness error will be negligible for the systems studied. The correlation energies for the atomic systems are taken from the
Z. Is˘tvanková et al. / Computational and Theoretical Chemistry 1067 (2015) 64–70
Fig. 3. The change in correlation energy, DENcorr ¼ ENcorr EN1 corr , for the N ¼ 3 and 4 (a) and, the N = 5–10 (b) series as a function of Z.
summary given by Chakravorty et al. [6,7] Throughout this paper the subscripts 1, 2, 3, 4 and 5 refer to the 1s; 2s; 2px ; 2py and 2pz orbitals, respectively. 3. Modelling of the atomic systems 3.1. Expansion of the electron correlation as a function of J;
pffiffiffi T and hri
The behaviour of the correlation energy as a function of N and Z is illustrated in Figs. S4 and S5, respectively, for the 2–18 electron series (see supporting information). The total Coulomb energy (J), exchange energy (K), potential energy (V), square root of the pffiffiffi kinetic energy ( T ), as well as all energy terms pffiffiffiffiffiffiffi ðJ ab ; K ab ; V ab ; T ab Þ, are all linear functions of Z (Figs. S1–S3). The inverse of the average value of r (hri1 ) is also a linear function of pffiffiffi Z (Fig. S10). Using these relationships, the substitution of J; T and hri for Z in the correlation energy expansion in Z (based on Eq. (2)) [6] results in: Model 1
Fig. 4. X 11 as a function of Z (a) and the relationship between Ecorr and X 11 (b) for the N ¼ 2 series.
coefficients resulted in a RMSD of 102 . Similar expansions in terms of hri are given in Table 2. In that case the N = 4–6 isoelectronic series have a significant contribution from the hri1 term. It is interesting to note that, for systems with low charge, the correlation energy P correlates well with a J aa where, J aa is the Coulomb repulsion between electrons in orbital a and the sum is over all doubly occupied orbitals. Deviations start occurring for ions with charges greater than five (Fig. S9). 3.2. Extending the Liu–Parr model The linear dependence of J on Z could also be used to modify the correlation energy model described by Liu and Parr [17]. For the H-atom and one-electron ions, the density at the nucleus, qð0Þ, is related to Z as:
qð0Þ / Z 3
ð11Þ 2
Furthermore, for all systems studied, we found Nqð0Þ=Z to be linearly related to J. Based on these relationships the Liu–Parr model, Eq. (5), can be expressed as, 3cLP
Ecorr ðN; JÞ ¼ B1 J 1 þ B0 J 0 þ B1 J 1 þ B2 J2 pffiffiffi pffiffiffi1 pffiffiffi0 pffiffiffi1 pffiffiffi2 Ecorr ðN; T Þ ¼ C 1 T þ C 0 T þ C 1 T þ C 2 T Ecorr ðN; hriÞ ¼ D2 hri2 þ D1 hri1 þ D0 hri0 þ D1 hri1
ð8Þ
Ecorr ¼ C LP ðaJ J þ bJ Þ
ð9Þ ð10Þ
pffiffiffi One advantage of the expansion of the correlation energy in J; T or hri over the expansion in Z is that it could also be applicable to modelling correlation energy of molecular systems. The optimized coefficients for the truncated correlation energy expansion in Z are summarized and compared to the coefficients in the correlation pffiffiffi energy expansions in J and T in Table 1. As expected, the coefficients Ai ; Bi and C i ði ¼ 1; 2; 3; 4Þ behave similarly with a change in N. For all three cases there is a much larger contribution from the pffiffiffi first order term ðZ; J; T Þ for the N ¼ 4; 5 and 6 isoelectronic series. Attempts to fit the N = 2–10 electron series with a single set of
67
Fig. 5. X 11 as a function of Z for the N = 2–10 series.
ð12Þ
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68
We found that Eq. (12) can be simplified without affecting the quality of the fits to, Model 2
Ecorr ¼ C 2 J c2
ð13Þ
As seen in Table 3 the parameters obtained using the pVTZ basis set are similar to those obtained by Liu and Parr [17]. In both the Liu– Parr model and Model 2 the parameters depend on N. Similar trends in the behaviour of the c2 and cLP parameters (Figs. S6 and S7) and the C 2 and C LP parameters (Fig. S8) with the increasing number of electrons suggest that these models are inter-related due to the relationship between Z and J. Model 2, a function of the total Coulomb energy only, performs as well as the Parr–Liu model. The relationship between the coefficients in the above models and N follows a similar trend. Although, the coefficients Ai ; Bi pffiffiffi and C i in the correlation energy expansion in Z; J and T , respectively, and the parameters C j and cj in Model 2 and in the Liu–Parr model could be approximated for all N, the extrapolation implemented this way would not lead to accurate results. For each model the parameters exhibit irregular behaviour with respect to N. The correlation energy is not expected to be a smooth function of N, although, it likely behaves smoothly in certain N intervals, such as for N = 5–7, N = 7–10, or for N = 13–15 and N = 15–18, for example (Fig. S4).
series) and very little change when electrons are paired. This would suggest that for Model 2 the total Coulomb repulsion in Eq. (13) could be replaced by JsN . As seen in the supporting information such a model performs equally well. 3.3. Correlation energy as a function of Coulomb and kinetic energies Rassolov [19] proposed a linear correlation operator which approximates the correlation energy for the single determinantal wavefunction based on its dependence on the separation of two electrons and their relative momentum (Eq. (7)). Starting with Rassolov’s operator, it is possible to devise other simple electron correlation models based on the properties of HF orbitals and the total wavefunction. Replacing the expectation values of the inter-electronic separation r212 and the relative momentum p212 operators in Rassolov’s correlation operator by the average inter-particle distance [12] and by the average inter-particle momentum, respectively, we arrive at:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi habjr 212 jabi dr ab qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ habjp212 jabi dpab
ðdr12 Þab ¼
ð16Þ
ðdp12 Þab
ð17Þ
and
Eab corr ¼ 3.2.1. The behaviour of qð0Þ and Coulomb repulsion As shown in Fig. 1 it is interesting to note that N
N
Dqð0Þ ¼ qð0Þ qð0Þ
N1
ð14Þ
increases for N ¼ 3 and 4 (Fig. 1(a)) but decreases for N = 6–10 (Fig. 1(b)). For N ¼ 5; Dqð0Þ5 initially decreases and starts increasing at higher values of Z. There is a clear distinction when adding electrons to s orbitals, which causes an increase in qð0Þ and adding electrons to p orbitals, which causes a decrease in qð0Þ. Because of its relationship with the density at the nucleus, a similar behaviour is expected for the change in Coulomb repulsion. The change in the total Coulomb repulsion associated with only the 1s and 2s electrons is defined as,
DJsN ¼ JsN JsN1
ð15Þ
As seen in Fig. 2(a) change in density at the nucleus does indeed translate to a corresponding change in JsN . In that case DJsN increases linearly for N ¼ 3 and N ¼ 4 (Fig. 2(a)) and decreases for N = 5 to N = 10 (Fig. 2(b)). The equivalent plots for the change in Ecorr are shown in Fig. 3. Fig. 3(a) clearly shows the large (and linear) increase in electron correlation when filling the 2s orbital (N ¼ 4 series). Adding electrons to the 2p orbitals (Fig. 3(b)), on the other hand, causes a decrease in electron correlation (N = 5, 6 and 7
C ab ðdr 12 Þab ðdp12 Þab
ð18Þ
Poirier and Hollett [12] observed that the average distance between two electrons, one in occupied molecular orbital a and one in occupied molecular orbital b; ðdr12 Þab is inversely proportional to the two-electron Coulomb integral,
ðdr12 Þab ¼
a
ð19Þ
J ab
Using the above relationship for the average inter-particle distance and a similar expression for the average inter-particle momentum, with me ¼ 1 a:u:,
ðdp12 Þab ¼
pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T aa þ T bb
ð20Þ
The correlation energy for a given electron pair associated with molecular orbitals a and b becomes:
J ab Eab corr ¼ C ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T aa þ T bb
ð21Þ
The total correlation energy can be written as a sum of the pair correlation energies:
Ecorr ¼
X J ab C ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T aa þ T bb ab
ð22Þ
Table 4 The optimized coefficients and the RMSD (in a.u.) for Models 3, 4 and 5. N
2 3 4 5 6 7 8 9 10 a b c
Eq. (25). Eq. (26). Eq. (27).
Model 3a
Model 4b
Model 5c
aX11 3
RMSD
aX11 4
aJ11 4
RMSD
aX11 5
aX22 5
aJ12 5
RMSD
0.0734 0.0825 0.4049 0.3669 0.3483 0.3435 0.4487 0.5474 0.6423
7.3E04 1.6E03 8.9E02 5.5E02 2.9E02 9.7E03 9.3E03 8.2E03 6.2E03
0.0734 0.0781 0.0945 0.1615 0.2318 0.3041 0.4069 0.5070 0.6090
0.0001 0.0003 0.0198 0.0127 0.0070 0.0023 0.0024 0.0022 0.0018
4.4E04 8.9E04 2.7E03 3.8E03 4.0E03 3.5E03 2.6E03 1.7E03 9.3E04
0.0734 0.0784 0.3299 0.6903 0.9711 1.1701 1.2783 1.3996 0.8936
– – 0.4002 0.9774 1.3993 1.6624 1.6765 1.7242 0.5439
– 0.0004 0.0141 0.0083 0.0037 0.0001 0.0003 0.0004 0.0009
7.3E04 8.8E04 2.8E04 6.1E05 1.4E04 8.4E05 2.5E04 2.3E04 8.0E04
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Table 5 The optimized parameters and RMSD (in a.u.) for Model 6.a N
2 3 4 5 6 7 8 9 10 a b c
Coefficients
RMSD
aX11 6
aX12 6
aX22 6
aX13 6
aX23 6
b aX34 6
0.0734 0.0549 0.3180 0.5499 1.0589 2.1462 2.6022 3.2960 3.4503
– 0.0341 0.1463 0.0792 0.1900 0.7531 0.8195 0.9523 0.8497
– – 0.4990 1.1432 0.8581 0.2644 0.0 0.0064 0.0
– – – 0.0861 0.2535 0.4622 0.2725 0.2355 0.1551
– – – 0.1816 0.3076 0.0872 0.4181 0.4980 0.4997
– – – – 2.0651 1.1941 0.0294 0.0614 0.1329
c aX33 6
aK6 – –
– 0.0 0.0 0.2321 0.0052 0.0088
– 0.0018 0.2679 0.1595 0.0718 0.0003 0.0012 0.0013 0.0003
7.3E04 1.2E04 1.7E05 1.9E05 6.9E06 3.2E05 7.5E05 5.2E05 2.7E05
Eq. (28). X 34 ¼ X 35 ¼ X 45 . X 33 ¼ X 44 ¼ X 55 .
where C ab would differ from the value of C 0R in the Rassolov operator, and the parameter C ab depends on the number of electrons in the system. For simplicity, a new variable X ab is defined as,
J ab X ab ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T aa þ T bb
ð23Þ
and the relationship between the correlation energy and the variable X ab is examined. For the two-electron series,
Ecorr / X 11
ð24Þ
In this case, the behaviour of X 11 with increasing Z (Fig. 4(a)) resembles that of the correlation energy with the increasing Z (Fig. S5). The relation between Ecorr and X 11 is also shown in Fig. 4(b). Interestingly, X 11 for the other isoelectronic series (Fig. 5) also resembles the behaviour of the correlation energy with increasing nuclear charge, except for the atomic series containing four, five, and six electrons. It was therefore initially investigated whether the total correlation energy for the atomic systems could be approximated using only X 11 . Model 3
Ecorr ¼ aX11 3 X 11
ð25Þ
Based on the Liu–Parr model, which suggests that most of the electron correlation can be accounted for by the core electrons, we tested two additional simple models. In these models only terms involving 1s and 2s orbitals are included and the intercept is taken to be zero. Model 4 J11 Ecorr ¼ aX11 4 X 11 þ a4 J 11
ð26Þ
Model 5 J12 X22 Ecorr ¼ aX11 5 X 11 þ a5 X 22 þ a5 J 12
ð27Þ
The resulting fits and the optimized parameters are summarized in Table 4. Model 3 does very poorly at describing the N = 4, 5 and 6 electron series. As expected, based on Table 1 and Eq. (8), adding the J11 term reduces the RMSD by an order of magnitude for the N = 4, 5 and 6 electron series. The RMSD is further reduced for Model 5. This simple model with only three parameters performs well for all the isoelectronic series. This is consistent with the trends observed in the change in electron correlation (Fig. 3) and with the change in the density at the nucleus, Fig. 1. For all cases Model 5 recovers on average 100% of the correlation energy. The correlation energy is overestimated (107%) or underestimated (98%) only for some of the N ¼ 2 and N ¼ 3 ions. To determine whether there is any advantage to be gained by increasing the number of empirical parameters we investigated the following model,
Model 6
Ecorr ¼
X K aXab 6 X ab þ a6 K 12
ð28Þ
ab
The correlation energy is expressed as a sum of the pair correlation energies, where each electron pair is characterized by a specific value of X ab and a K 12 term. For Model 6 all coefficients are allowed to vary independently. The results are summarized in Table 5. Although Model 6 is successful in predicting accurate correlation energies with lower RMSDs it provides very little new insight. 4. Conclusions A total of six different models were investigated for the N = 2– 10 isoelectronic series. The performance of the correlation energy models for the atomic systems was assessed using the RMSD. Many of the models presented here can successfully account for the missing correlation energy in the atomic isoelectronic series. Analysis of DENcorr ; DqN ð0Þ and DJ sN shows that there is a clear distinction between filling the 2s orbital and the 2p orbitals. Filling the 2s orbital causes an increase in electron correlation while adding unpaired electrons to the 2p orbitals (N = 5, 6, 7) causes a decrease in electron correlation. On the other hand, when further electrons are added to the 2p orbitals the change in correlation energy is small (N = 8, 9, 10). This provides a new perspective to the different behaviour of the N = 4 to N = 6 electron series in all the models. Model 5, with only three terms (X 11 ; X 22 and J 12 ) performed consistently well with the RMSD ranging from 9 104 (N ¼ 3) to 6 105 (N ¼ 5). Consistent with Rassolov’s linear corpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relation operator [19], X ab ¼ J ab = T aa þ T bb , especially where, a ¼ b, can account for the behaviour of electron correlation in these isoelectronic series. The advantages of the correlation models presented here is their computational efficiency and possible applicability to a wide range of larger chemical system. Acknowledgements We gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Memorial University of Newfoundland, Compute Canada and the Atlantic Excellence Network (ACEnet), regional high performance computing consortium for universities in Atlantic Canada. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.comptc.2015.04. 015.
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