The Coulomb, exchange, and correlation components of the electron-electron repulsion in harmonium atoms Jerzy Cioslowski
Citation: The Journal of Chemical Physics 142, 114105 (2015); doi: 10.1063/1.4914021 View online: http://dx.doi.org/10.1063/1.4914021 View Table of Contents: http://aip.scitation.org/toc/jcp/142/11 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 142, 114105 (2015)
The Coulomb, exchange, and correlation components of the electron-electron repulsion in harmonium atoms Jerzy Cioslowski Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
(Received 28 January 2015; accepted 23 February 2015; published online 17 March 2015) Highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the three-electron harmonium atoms in the 2 P− and 4 P+ states are obtained for 19 values of the confinement strength ω ranging from 10−3 to 103. The computed data are consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to these limits are constructed, opening an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of the dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are expected to be particularly useful in the context of benchmarking of approximate density matrix functionals. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914021]
I. INTRODUCTION
There exist physical systems for which the time-independent Schödinger equation can be solved only for particular values (including limits) of a certain parameter that enters the pertinent nonrelativistic Hamiltonian. Such quasi-solvable systems play a dual role in development of electronic structure methods. First, they serve as benchmarking tools that make it possible to assess the accuracy of approximate approaches of quantum chemistry without resorting to comparisons with experimental data, which are often prone to errors acquired at both the actual measurement and the subsequent adjustments (such as e.g., the corrections for relativistic effects and anharmonicities). Second, they provide sets of constraints facilitating construction of approximate functionals in methods, such as the density and density matrix functional theories (DFT and DMFT, respectively), that dispose of the electronic wavefunction as the fundamental quantity. At present, two quasi-solvable model systems are widely employed in these roles. The first of them, namely, the homogeneous electron gas (HEG), i.e., an assembly of infinitely many electrons with a constant spatial density immersed in a uniform charge-compensating background, usually serves as the point of departure in derivations of energy functionals of DFT.1 While benefiting from mathematical simplicity and the availability of closed-form asymptotic expressions for energies and other properties at both the lowand high-density limits (augmented with relatively accurate data for finite densities obtained mostly from numerical Monte Carlo simulations2,3), it suffers from limited relevance to highly inhomogeneous species such as atoms, molecules, and extended systems. Consequently, some approaches, such as the local-density approximation of DFT, that are quite accurate for the HEG perform poorly for atoms and molecules, whereas others, such as those based upon the JK-only approximation of 0021-9606/2015/142(11)/114105/7/$30.00
DMFT, fare reasonably well in actual calculations4–6 despite blatantly failing the HEG test.7 The other model system of importance in this context is the harmonium atom, i.e., the species described by the Hamiltonian N N 1 ˆ2 1 Hˆ = (−∇i + ω2 r i2) + . 2 i=1 r i > j=1 i j
(1)
Electronic properties of this model atom have been thoroughly studied with both rigorous mathematical analysis8–18 and numerical simulations.19–22 For large values of the confinement strength ω, the harmonium atoms closely resemble their ordinary counterparts, whereas at the strong-correlation limit of ω → 0, they become archetypes of species with exclusively nondynamical correlation and complete spatial localization of electrons (i.e., Wigner crystallization).8–10,14,16,18 This tunability of the relative extents of the dynamical and nondynamical electron correlation effects, combined with the availability of exact wavefunctions and energies for certain 1 values of ω (such as 12 , 10 , etc.), has resulted in employment of the two-electron harmonium atom as a benchmarking tool for both the DFT-based23–29 and other30–32 formalisms of quantum chemistry. The Hamiltonians of harmonium atoms comprising more than two electrons do not possess eigenfunctions that are reducible to closed forms except at the ω → 0 and ω → ∞ limits. However, thanks to the abundance of electronic states with diverse spins and character (singly or multideterminantal),17,21 such model systems provide an even richer, though largely unexplored, testing ground for approximate approaches to the electron correlation problem. Moreover, in contrast to the N = 2 case, their usefulness is not diminished by the exactness of many electronic structure methods for two-electron systems.
142, 114105-1
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J. Chem. Phys. 142, 114105 (2015)
In Coulombic systems, partitioning of the electronelectron repulsion energy W parallels that of the 2-matrix.33,34 Thus, W consists of the direct (Coulomb) and exchange components given by the expressions J=
1 2
W (Q; ω) =
∞
j E ( j)(Q) ω(2− j)/2,
(5)
j=0
Γ(⃗r 1,⃗r 1) Γ(⃗r 2,⃗r 2) |⃗r 1 − ⃗r 2|−1 d⃗r 1 d⃗r 2
(2)
and 1 K =− 2
Application of a simple scaling argument leads to the conclusion that
Γ(⃗r 1,⃗r 2) Γ(⃗r 2,⃗r 1) |⃗r 1 − ⃗r 2|−1 d⃗r 1 d⃗r 2,
(3)
i.e., the jth-order contribution W ( j)(Q) to the electron-electron repulsion energy W (Q; ω) equals j times the corresponding contribution E ( j)(Q) to the total energy E(Q; ω). For a state Q that tends to a single Slater determinant Ψ0 at the limit of ω → ∞, one readily obtains E (1)(Q) = W (1)(Q) =
′
respectively, that involve the spin-summed 1-matrix Γ(⃗r 1, r⃗1 ) and the remainder U due to electron correlation. Further partitioning of J and K yields the respective spin-pair contributions Jαα , Jα β , Jβα , Jβ β , Kαα , and K β β . In DMFT, U is the only energy component for which the explicit functional is not known (except for the singlet ground states of two-electron systems). Therefore, the values of U paired with the corresponding 1-matrices constitute an ideal development and benchmarking tool for approximate DMFT approaches. Prompted by this observation, exhaustive computations of the electron-electron repulsion energy components have been recently completed for the two lowest-energy 2 P− and 4 P+ states of the three-electron harmonium atom. Their results are presented in this paper. II. THEORY AND DETAILS OF CALCULATIONS
In order to attain the ultimate objective of the present work, i.e., derivation of formulae capable of yielding highly accurate estimates of the electron-electron repulsion energy components for arbitrary ω, the weak- and strong-correlation asymptotics of these quantities have to be obtained and then combined with numerical data computed at several values of the confinement strength.
1 ⟨ab∥ab⟩ 2 ab
(6)
and E (2)(Q) =
2 1 (2) W (Q) = − (E p − E a )−1 ⟨ab∥pb⟩ 2 b ap 2 1 − (E p + E q − E a − E b )−1 ⟨ab∥pq⟩ , 4 ab pq (7)
where E k is the energy of the kth spinorbital of a threedimensional harmonic oscillator (with unit mass and force constant) and ⟨i j ∥kl⟩ ≡ ⟨i j|kl⟩ − ⟨i j|l k⟩ is the antisymmetrized variant of the two-electron integral ⟨i j|kl⟩ ≡ ⟨i(1) −1 j(2)|r 12 |k(1)l(2)⟩ involving the spinorbitals with the indices i, j, k, and l. Here and in the following, a, b, and c are the indices of the spinorbitals included in Ψ0, whereas p and q denote those belonging to their orthogonal complement (i.e., the virtual spinorbitals). The respective first-order contributions, J (1)(Q) =
1 ⟨ab|ab⟩ 2 ab
(8)
and A. The weak-correlation limit
At the ω → ∞ (strong-confinement or weak-correlation) limit, the total energy E(Q; ω) of a given electronic state Q of a harmonium atom possesses the asymptotics9,11–13,15,17 E(Q; ω) =
∞
E ( j)(Q) ω(2− j)/2.
(4)
j=0
K (1)(Q) = −
1 ⟨ab|ba⟩, 2 ab
(9)
to the Coulomb and exchange components of W (Q; ω) [J(Q; ω) and K(Q; ω), respectively] are trivial to compute. On the other hand, derivation of the second-order expressions, J (2)(Q) = −2 Re (E p − E a )−1 ⟨ab|pb⟩ ⟨pc∥ac⟩ (10) abc p
At present, closed-form expressions for the coefficients E ( j)(Q) with 0 ≤ j ≤ 2 are known for the 1 S+ and 3 P− electronic states of the two-electron harmonium atom,9,11–13,15,17 the 2 P− and 4 P+ states of the three-electron species,15,17 and the 3 P+, 1 D+, 1 S+, and 5 S− states of the four-electron system.17 In addition, the coefficient E (3)(1 S+) is available in a closed form for the two-electron harmonium atom.12
Jα(2)β (Q) = −Re
and K (2)(Q) = 2 Re
(E p − E a )−1 ⟨ab|bp⟩ ⟨pc∥ac⟩,
(11)
abc p
is somewhat more involved. The expressions for the individual spin-pair components, such as
⟨a β bα |pβ bα ⟩ ⟨pβ c∥a β c⟩ + ⟨aα bβ |pα bβ ⟩ ⟨pα c∥aα c⟩ , E p − Ea abc p
(12)
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TABLE I. The first-order contributions to the components of the electronelectron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom at the weak-correlation limit.a
K (1)(Q)
Q = 2 P− √ (209/120) 2/π √ (1/2) 2/π √ (11/6) 2/π √ (163/40) 2/π √ −(43/40) 2/π √ −(1/2) 2/π √ −(63/40) 2/π
U(1)(Q)
0
Component (1)
Jαα (Q) (1)
J β β (Q) (1) Jα β (Q) J(1)(Q) (1)
K αα (Q) (1)
K β β (Q)
a See
Computation of the sums that enter Eqs. (8)–(13) involves the previously published algebraic approaches15,17 that employ summation techniques based upon properties of hypergeometric functions.35 All the second-order quantities are reducible to the standard form √ 3 1 ln 2 F (C1,C2,C3,C4,C5) = C1 + C2 + C3 + C4 π π π √ ln(1 + 3) C5, (14) + π where the coefficients {C1,C2,C3,C4,C5} are rational numbers.
Q = 4 P+ √ (37/10) 2/π 0 0 √ (37/10) 2/π √ −(17/10) 2/π 0 √ −(17/10) 2/π
B. The strong-correlation limit
0
At the ω → 0 (weak-confinement or strong-correlation) limit, E(Q; ω) possesses the asymptotics9,14,16,20,21
(1)
Eqs. (8) and (9); the entries labeled Jα β (Q) actually list the sums of two equal (1)
(1)
contributions Jα β (Q) and J βα (Q). The spin-summed components are boldfaced.
E(Q; ω) =
2 1 (E p + E q − E a − E b )−1 ⟨ab∥pq⟩ , 2 ab pq
E˜ ( j)(Q) ω(2+ j)/3.
Analogous power series describe the asymptotic behavior of the electron-electron repulsion energy [where a simple scaling argument yields W˜ ( j)(Q) = 32 (1 − j) E˜ ( j)(Q)] and its Coulomb component.18 The exchange component of the electron-electron repulsion energy also possesses the small-ω power expansion
(13)
K(Q; ω) =
∞
K˜ ( j)(Q) ω(5+2 j)/6.
The expansions for the individual spin-pair components [i.e., Jαα (Q; ω), Jα β (Q; ω), Jβα (Q; ω), Jβ β (Q; ω), Kαα (Q; ω), and K β β (Q; ω)] are analogous to those for their spin-summed counterparts.
TABLE II. The second-order contributions to the components of the electron-electron repulsion energies of the 2 P and 4 P states of the three-electron harmonium atom at the weak correlation limit.a − + C1
C2
C3
C4
C5
Numerical value
(2) Jαα (2 P−)
208/27
−6467/270
23/45
−1991/30
1991/45
−0.126 865 317
(2) J β β (2 P−) (2) Jα β (2 P−)
8/3
−8
0
−22
44/3
−0.041 662 624
1216/135
−3731/135
16/45
−226/3
452/9
−0.147 920 981
872/45
−5363/90
13/15
−1637/10
1637/15
−0.316 448 922
J(2)(2P−) (2) K αα (2 P−)
−688/135
815/54
1/9
1231/30
−1231/45
0.070 959 049
K β β (2 P−)
−8/3
8
0
22
−44/3
0.041 662 624
K (2)(2P−)
−1048/135
1247/54
1/9
1891/30
−1891/45
0.112 621 673
U(2)(2P−)
−98/135
967/135
−14/45
43
−310/9
−0.149 481 001
Jαα (4 P+)
592/45
−2096/45
64/15
−592/5
1184/15
−0.189 323 324
(2) J β β (4 P+) (2) 4 J α β ( P +) J(2)(4P+)
0
0
0
0
0
0
0
0
0
0
(2)
(2)
592/45
−2096/45
64/15
−592/5
1184/15
−0.189 323 324
K αα (4 P+)
−368/45
944/45
32/15
272/5
−544/15
0.076 035 997
(2) K β β (4 P+) K (2)(4P+)
0
0
0
0
0
−368/45
944/45
32/15
272/5
−544/15
0.076 035 997
2/15
64/15
−16/15
80/3
−64/3
−0.037 933 367
(2)
U(2)(4P+) a See
Eqs. (10)–(14); the entries labeled spin-summed components are boldfaced.
(2) Jα β (Q)
(16)
j=0
i.e., the second-order contribution U (2)(Q) to the correlation component U(Q; ω) of W (Q; ω) equals twice the second term in the r.h.s. of Eq. (7) or, in other words, twice the second-order contribution to the correlation energy.
Component
(15)
j=0
where the subscripts α and β indicate spin restrictions, are obtained in a similar fashion. The first term in the r.h.s. of Eq. (7), which accounts for the second-order contribution to the Hartree-Fock energy, is readily shown to equal one-half of the sum J (2)(Q) + K (2)(Q). Consequently, U (2)(Q) = −
∞
actually list the sums of two equal contributions
(2) Jα β (Q)
(2)
and J βα (Q). The
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J. Chem. Phys. 142, 114105 (2015)
TABLE III. The leading coefficients of the small-ω expansions for the Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Q = 2 P−
Q = 4 P+
(0) J˜αα (Q)
(17/8) 31/6
(117/25) 31/6
(0) J˜β β (Q)
(13/25) 31/6
0
(0) J˜α β (Q) J˜ (0)(Q)
(19/10) 31/6
0
(909/200) 31/6
(117/25) 31/6
(0) K˜ αα (Q)
−2.335 685 308
−4.826 270 917
(0) K˜ β β (Q)
−0.536 252 324
0
Component
K˜ (0)(Q)
−2.871 937 632
−4.826 270 917
˜ (0)(Q) U
32/3 − (909/200) 31/6
32/3 − (117/25) 31/6
˜ (1)(Q) U
2.871 937 632
4.826 270 917
a See
Eqs. (15)–(17); the entries labeled (0)
(0) J˜α β (Q)
actually list the sums of two equal
(0)
contributions J˜α β (Q) and J˜βα (Q). The spin-summed components are boldfaced.
The small-ω behavior of the correlation component U(Q; ω) is described by yet another series, i.e., U(Q; ω) =
∞
U˜ ( j)(Q) ω(4+ j)/6,
(17)
j=0
with U˜ (0)(Q) = W˜ (0)(Q) − J˜(0)(Q) and U˜ (1)(Q) = −K˜ (0)(Q) [note the mismatch in the powers of ω that enter Eqs. (15) and (16)].
are obtained from Eqs. (2) and (3) by employing finite-basis representations of the pertinent 1-matrices. Computation of the relevant matrix elements involving the basis functions φ I (r) = ζ 3/4 H I x ( ζ x)H I y ( ζ y)H Iz ( ζ z) exp(−ζ r 2/2), (18) where Hk (x) is the kth normalized Hermite polynomial and I ≡ (I x , I y , Iz ), has been described elsewhere.20 In practice, limiting this basis set to the functions with 0 ≤ I x , I y , Iz ≤ 10 is found to produce sufficiently accurate data provided the exponent ζ is optimized de novo for each value of ω. Two-electron integrals involving these functions, which are necessary for evaluations of the r.h.s. of Eqs. (2) and (3), are readily available from the algorithm of McMurchie and Davidson.36 The correlation components are computed simply as the differences between the electron-electron repulsion energies and the sums of their Coulomb and exchange components. The recently introduced robust interpolation between weak- and strong-correlation regimes of quantum systems37 provides a recipe for construction of approximants that accurately reproduce exact values of observables while conforming to their asymptotics at both limits. In the present case, M known approximants for the Coulomb components of the the 0+1 electron-electron repulsion energy are given by J(Q; ω) =
M
CM,k (1 − t) M−k−1 t k+2,
(19)
k=0
where t is the real-valued solution of the equation ω = ω0 (1 − t)−2 t 3
C. Numerical calculations for finite confinement strengths
For finite confinement strengths, the Coulomb and exchange components of the electron-electron repulsion energy
with a parameter ω0. The coefficients CM,0, CM, M , and CM, M−1 are derived from J˜(0)(Q), J (1)(Q), and J (2)(Q) (or their spinpair components), and ω0, which in turn is obtained (together
TABLE IV. The Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the 2 P− ground state of the three-electron harmonium atom.a ω 1000 500 200 100 50 20 10 5 2 1 0.5 0.2 0.1 0.05 0.02 0.01 0.005 0.002 0.001 a The
(20)
Jαα (2 P−;ω)
J β β (2 P−;ω)
Jα β (2 P−;ω)
K αα (2 P−;ω)
K β β (2 P−;ω)
U (2 P−;ω)
43.818 271 30.947 451 19.527 037 13.771 460 9.702 027 6.091 906 4.273 310 2.988 487 1.850 763 1.279 800 0.878 864 0.528 163 0.355 728 0.237 563 0.137 631 0.090 309 0.058 882 0.033 186 0.021 406
12.574 127 8.879 137 5.600 517 3.948 162 2.779 855 1.743 365 1.221 184 0.852 216 0.525 408 0.361 351 0.246 150 0.145 534 0.096 345 0.063 032 0.035 521 0.022 915 0.014 766 0.008 249 0.005 302
46.110 083 32.561 832 20.540 312 14.481 773 10.198 111 6.397 898 4.483 501 3.130 961 1.933 240 1.332 196 0.910 261 0.541 655 0.361 059 0.238 110 0.135 470 0.087 764 0.056 612 0.031 557 0.020 220
−27.052 493 −19.108 066 −12.058 639 −8.505 637 −5.993 165 −3.763 531 −2.639 567 −1.844 572 −1.138 840 −0.783 083 −0.531 831 −0.310 426 −0.201 188 −0.127 025 −0.066 520 −0.039 801 −0.023 459 −0.011 481 −0.006 630
−12.573 700 −8.878 536 −5.599 573 −3.946 837 −2.778 000 −1.740 492 −1.217 217 −0.846 796 −0.517 422 −0.350 950 −0.233 104 −0.129 361 −0.078 915 −0.045 919 −0.021 169 −0.011 569 −0.006 357 −0.002 932 −0.001 647
−0.149 017 −0.148 825 −0.148 444 −0.148 016 −0.147 411 −0.146 217 −0.144 875 −0.142 986 −0.139 254 −0.135 070 −0.129 209 −0.117 926 −0.106 122 −0.091 573 −0.069 922 −0.053 929 −0.039 954 −0.025 655 −0.017 886
entries labeled Jα β (2 P−; ω) actually list the sums of two equal contributions Jα β (2 P−; ω) and J βα (2 P−; ω).
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TABLE V. The Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the 4 P+ first excited state of the three-electron harmonium atom. ω 1000 500 200 100 50 20 10 5 2 1 0.5 0.2 0.1 0.05 0.02 0.01 0.005 0.002 0.001
Jαα (4 P+; ω)
K αα (4 P+; ω)
U(4 P+; ω)
93.167 137 65.824 057 41.561 951 29.334 165 20.688 175 13.017 104 9.151 765 6.419 651 3.997 682 2.779 507 1.921 101 1.165 219 0.789 781 0.529 862 0.307 761 0.201 901 0.131 493 0.073 984 0.047 679
−42.817 185 −30.254 031 −19.106 343 −13.487 913 −9.515 069 −5.989 820 −4.213 085 −2.956 743 −1.842 008 −1.280 323 −0.883 451 −0.532 229 −0.356 530 −0.234 138 −0.129 360 −0.080 045 −0.048 273 −0.023 974 −0.013 916
−0.037 935 −0.037 936 −0.037 937 −0.037 938 −0.037 939 −0.037 937 −0.037 930 −0.037 913 −0.037 851 −0.037 742 −0.037 519 −0.036 885 −0.035 942 −0.034 368 −0.030 989 −0.027 392 −0.023 119 −0.017 168 −0.013 023
with the remaining coefficients {CM,k }) by minimizing the differences between the values of J(Q; ω) computed at 19 confinement strengths ranging from 10−3 to 103 20 and their respective estimates obtained from Eqs. (19) and (20). The M construction of the 0+1 approximants for the exchange components of the electron-electron repulsion energy proceeds analogously, with K˜ (0)(Q), K (1)(Q), and K (2)(Q) serving as the asymptotic constraints, and K(Q; ω) =
M
CM,k (1 − t) M −k−1 t k+5/2
(21)
k=0
replacing Eq. (19). The asymptotic data employed for the correlation compo(2) ˜ (0) ˜ (1) nent M consist of U (Q), U (Q), and U (Q), affording the 1+0 approximants
U(Q; ω) =
M
CM,k (1 − t) M −k t k+4,
where t is the real-valued solution of the equation ω = ω0 (1 − t)−2 t 6.
ω0 C 9,0 C 9,1 C 9,2 C 9,3 C 9,4 C 9,5 C 9,6 C 9,7 C 9,8 C 9,9
III. RESULTS AND DISCUSSION
The first- and second-order contributions to the Coulomb, exchange, and correlation components of the electron-electron repulsion energies of weakly correlated three-electron harmonium atoms in the 2 P− and 4 P+ states are compiled in Tables I and II. As expected, the first-order contributions U (1)(Q) to the correlation components equal zero and their second-order counterparts U (2)(Q) equal twice the asymptotic values of the respective correlation energies that have been obtained previously.15,17 In the doublet ground state, there is a perfect cancellation of the Jβ β (Q; ω) and K β β (Q; ω) components at both orders. At the ω → ∞ limit, the J(Q; ω)/W (Q; ω) ratios 37 for the 2 P− and 4 P+ states tend to 163 100 and 20 , respectively. The corresponding ratios for the exchange components approach 63 − 100 and − 17 20 , whereas those for the correlation components vanish. An entirely different picture emerges at the strong-correlation limit. With their leading asymptotics of K˜ (0)(Q) ω5/6, where the coefficients K˜ (0)(Q) are given by rather complicated expressions,18 the exchange components now become vanishingly small in comparison to their Coulomb and correlation counterparts. Inspection of Table III reveals the large asymp√ √ 39 totic values ( 303 3 ≈ 2.624 and 3 ≈ 2.702 for the 2 P− 200 25 4 and P+ states, respectively) of the J(Q; ω)/W (Q; ω) ratios at ω → 0 that indicate the presence of strong electron correlation effects in three-electron harmonium atoms within the weakconfinement regime.
Jαα (2 P−; ω)
J β β (2 P−; ω)
Jα β (2 P−; ω)
Jαα (4 P+; ω)
0.120 165 477 3 0.621 439 642 4 4.697 842 809 20.719 085 78 44.592 416 70 67.420 189 40 66.262 989 58 43.473 857 06 18.233 810 33 4.449 477 652 0.481 720 312 5
0.013 758 377 00 0.035 856 538 08 0.310 490 678 2 1.167 362 000 3.002 302 178 4.382 920 766 4.434 401 749 3.221 902 289 1.506 790 895 0.402 883 956 1 0.046 794 376 89
0.006 884 684 572 0.082 577 774 64 0.790 857 833 5 2.619 965 718 7.711 306 022 10.330 821 41 11.593 396 26 7.820 528 814 3.640 036 565 1.005 126 325 0.121 373 400 6
0.043 007 883 14 0.689 919 002 4 5.601 792 151 23.195 436 18 53.139 933 18 81.615 693 60 81.642 281 87 54.157 232 15 22.910 796 07 5.626 877 912 0.612 231 709 0
Eqs. (19) and (20); the entries labeled Jα β (2 P−; ω) actually list the parameters for the sum of two equal components Jα β (2 P−; ω) and J βα (2 P−; ω).
a See
(23)
Another, less convenient alternative involves subtracting the sum of the approximants for J(Q; ω) and K(Q; ω) [Eqs. (19)–(21)] from that for the electron-electron repulsion energy (which in turn is readily derived from the published approximant for the total energy20).
9 TABLE VI. The parameters of the 0+1 approximants for the Coulomb components of the electron-electron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Parameter
(22)
k=0
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J. Chem. Phys. 142, 114105 (2015)
9 approximants for the exchange components of the electron-electron TABLE VII. The parameters of the 0+1 repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Parameter ω0 C 9,0 C 9,1 C 9,2 C 9,3 C 9,4 C 9,5 C 9,6 C 9,7 C 9,8 C 9,9 a See
K αα (2 P−; ω)
K β β (2 P−; ω)
K αα (4 P+; ω)
0.081 252 946 53 −0.288 367 908 9 −2.669 235 156 −10.969 877 66 −24.633 118 17 −42.150 020 38 −40.308 258 33 −25.595 680 75 −10.232 089 29 −2.373 980 273 −0.244 493 932 1
0.007 125 965 535 −0.008 711 085 680 0.280 317 468 1 −2.208 794 758 2.770 510 883 −5.069 709 999 −0.154 998 307 2 −1.213 721 680 −0.964 374 841 6 −0.295 106 234 0 −0.033 676 885 83
0.015 346 671 11 −0.148 578 801 8 −1.650 428 016 −5.226 123 310 −14.971 359 03 −24.116 716 18 −24.822 273 96 −16.359 252 01 −6.755 137 165 −1.604 299 786 −0.168 033 578 3
Eqs.(20) and (21).
The computed values of the components of the electronelectron repulsion energies at finite confinement strengths are listed in Tables IV and V. As expected, the magnitudes of the correlation components decrease with the state multiplicity. This decrease is more pronounced within the strongconfinement regime which, coupled with the small difference in the respective ω → 0 asymptotic values (see Table III), results in a markedly reduced variation of U(Q; ω) with ω in the case of the 4 P+ state. On the other hand, weakening of the confinement brings about significantly faster decay of the exchange components relative to their Coulomb and correlation counterparts in both states under study. Whereas the aforediscussed energy components allow for assessing the accuracy of approximate electronic structure methods at 19 different values of ω, the approximants (19)–(23) provide the benchmarking data at arbitrary confinement strengths. At M = 9, these approximants faithfully repro 9 duce the data compiled in Tables IV and V. Thus, the 0+1 approximants with the parameters listed in Table VI deviate by at most 2.9, 0.6, 0.8, and 1.2 µhartree, respectively, from 2 the actual values of Jαα (2 P−; ω), Jβ β (2 P−; ω), 9Jα β ( P−; ω), and 4 Jαα ( P+; ω). The respective figures for the 0+1 approximants for Kαα (2 P−; ω), K β β (2 P−; ω), and Kαα (4 P+; ω) (Table VII) 9 are 2.1, 0.9, and 2.5 µhartree, whereas those for the 1+0
approximants for U(2 P−; ω) and U(4 P+; ω) (Table VIII) are 2.6 and 0.5 µhartree.
IV. CONCLUSIONS
The availability of highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of harmonium atoms with a wide range of confinement strengths opens an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are particularly useful in the context of benchmarking of approximate DMFT functionals. The results of such studies will be published elsewhere. The computed data are found to be consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Consequently, robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to those limits are readily constructed, further enhancing the utility of manyelectron harmonium atoms as model systems. ACKNOWLEDGMENTS
9 TABLE VIII. The parameters of the 1+0 approximants for the correlation components of the electron-electron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Parameter ω0 C 9,0 C 9,1 C 9,2 C 9,3 C 9,4 C 9,5 C 9,6 C 9,7 C 9,8 C 9,9 a See
Eqs. (22) and (23).
U(2 P−; ω)
U(4 P+; ω)
0.002 904 388 173 −0.068 767 811 52 −0.688 517 931 8 −124.378 520 9 247.823 608 9 −215.416 588 9 56.043 826 18 −16.901 055 23 −7.688 967 569 −1.671 063 414 −0.149 481 001 0
0.001 983 615 922 −0.055 891 435 04 −0.550 536 791 8 4.730 618 959 −21.398 513 44 3.960 747 515 −12.792 391 46 −8.588 516 527 −2.846 252 362 −0.494 850 754 8 −0.037 933 366 60
The research described in this publication has been funded by NCN (Poland) under Grant No. DEC-2012/07/B/ST4/ 00553. The support from MPI PKS Dresden is also acknowledged. 1E.
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Citation: The Journal of Chemical Physics 142, 114105 (2015); doi: 10.1063/1.4914021 View online: http://dx.doi.org/10.1063/1.4914021 View Table of Contents: http://aip.scitation.org/toc/jcp/142/11 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 142, 114105 (2015)
The Coulomb, exchange, and correlation components of the electron-electron repulsion in harmonium atoms Jerzy Cioslowski Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
(Received 28 January 2015; accepted 23 February 2015; published online 17 March 2015) Highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the three-electron harmonium atoms in the 2 P− and 4 P+ states are obtained for 19 values of the confinement strength ω ranging from 10−3 to 103. The computed data are consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to these limits are constructed, opening an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of the dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are expected to be particularly useful in the context of benchmarking of approximate density matrix functionals. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914021]
I. INTRODUCTION
There exist physical systems for which the time-independent Schödinger equation can be solved only for particular values (including limits) of a certain parameter that enters the pertinent nonrelativistic Hamiltonian. Such quasi-solvable systems play a dual role in development of electronic structure methods. First, they serve as benchmarking tools that make it possible to assess the accuracy of approximate approaches of quantum chemistry without resorting to comparisons with experimental data, which are often prone to errors acquired at both the actual measurement and the subsequent adjustments (such as e.g., the corrections for relativistic effects and anharmonicities). Second, they provide sets of constraints facilitating construction of approximate functionals in methods, such as the density and density matrix functional theories (DFT and DMFT, respectively), that dispose of the electronic wavefunction as the fundamental quantity. At present, two quasi-solvable model systems are widely employed in these roles. The first of them, namely, the homogeneous electron gas (HEG), i.e., an assembly of infinitely many electrons with a constant spatial density immersed in a uniform charge-compensating background, usually serves as the point of departure in derivations of energy functionals of DFT.1 While benefiting from mathematical simplicity and the availability of closed-form asymptotic expressions for energies and other properties at both the lowand high-density limits (augmented with relatively accurate data for finite densities obtained mostly from numerical Monte Carlo simulations2,3), it suffers from limited relevance to highly inhomogeneous species such as atoms, molecules, and extended systems. Consequently, some approaches, such as the local-density approximation of DFT, that are quite accurate for the HEG perform poorly for atoms and molecules, whereas others, such as those based upon the JK-only approximation of 0021-9606/2015/142(11)/114105/7/$30.00
DMFT, fare reasonably well in actual calculations4–6 despite blatantly failing the HEG test.7 The other model system of importance in this context is the harmonium atom, i.e., the species described by the Hamiltonian N N 1 ˆ2 1 Hˆ = (−∇i + ω2 r i2) + . 2 i=1 r i > j=1 i j
(1)
Electronic properties of this model atom have been thoroughly studied with both rigorous mathematical analysis8–18 and numerical simulations.19–22 For large values of the confinement strength ω, the harmonium atoms closely resemble their ordinary counterparts, whereas at the strong-correlation limit of ω → 0, they become archetypes of species with exclusively nondynamical correlation and complete spatial localization of electrons (i.e., Wigner crystallization).8–10,14,16,18 This tunability of the relative extents of the dynamical and nondynamical electron correlation effects, combined with the availability of exact wavefunctions and energies for certain 1 values of ω (such as 12 , 10 , etc.), has resulted in employment of the two-electron harmonium atom as a benchmarking tool for both the DFT-based23–29 and other30–32 formalisms of quantum chemistry. The Hamiltonians of harmonium atoms comprising more than two electrons do not possess eigenfunctions that are reducible to closed forms except at the ω → 0 and ω → ∞ limits. However, thanks to the abundance of electronic states with diverse spins and character (singly or multideterminantal),17,21 such model systems provide an even richer, though largely unexplored, testing ground for approximate approaches to the electron correlation problem. Moreover, in contrast to the N = 2 case, their usefulness is not diminished by the exactness of many electronic structure methods for two-electron systems.
142, 114105-1
© 2015 AIP Publishing LLC
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In Coulombic systems, partitioning of the electronelectron repulsion energy W parallels that of the 2-matrix.33,34 Thus, W consists of the direct (Coulomb) and exchange components given by the expressions J=
1 2
W (Q; ω) =
∞
j E ( j)(Q) ω(2− j)/2,
(5)
j=0
Γ(⃗r 1,⃗r 1) Γ(⃗r 2,⃗r 2) |⃗r 1 − ⃗r 2|−1 d⃗r 1 d⃗r 2
(2)
and 1 K =− 2
Application of a simple scaling argument leads to the conclusion that
Γ(⃗r 1,⃗r 2) Γ(⃗r 2,⃗r 1) |⃗r 1 − ⃗r 2|−1 d⃗r 1 d⃗r 2,
(3)
i.e., the jth-order contribution W ( j)(Q) to the electron-electron repulsion energy W (Q; ω) equals j times the corresponding contribution E ( j)(Q) to the total energy E(Q; ω). For a state Q that tends to a single Slater determinant Ψ0 at the limit of ω → ∞, one readily obtains E (1)(Q) = W (1)(Q) =
′
respectively, that involve the spin-summed 1-matrix Γ(⃗r 1, r⃗1 ) and the remainder U due to electron correlation. Further partitioning of J and K yields the respective spin-pair contributions Jαα , Jα β , Jβα , Jβ β , Kαα , and K β β . In DMFT, U is the only energy component for which the explicit functional is not known (except for the singlet ground states of two-electron systems). Therefore, the values of U paired with the corresponding 1-matrices constitute an ideal development and benchmarking tool for approximate DMFT approaches. Prompted by this observation, exhaustive computations of the electron-electron repulsion energy components have been recently completed for the two lowest-energy 2 P− and 4 P+ states of the three-electron harmonium atom. Their results are presented in this paper. II. THEORY AND DETAILS OF CALCULATIONS
In order to attain the ultimate objective of the present work, i.e., derivation of formulae capable of yielding highly accurate estimates of the electron-electron repulsion energy components for arbitrary ω, the weak- and strong-correlation asymptotics of these quantities have to be obtained and then combined with numerical data computed at several values of the confinement strength.
1 ⟨ab∥ab⟩ 2 ab
(6)
and E (2)(Q) =
2 1 (2) W (Q) = − (E p − E a )−1 ⟨ab∥pb⟩ 2 b ap 2 1 − (E p + E q − E a − E b )−1 ⟨ab∥pq⟩ , 4 ab pq (7)
where E k is the energy of the kth spinorbital of a threedimensional harmonic oscillator (with unit mass and force constant) and ⟨i j ∥kl⟩ ≡ ⟨i j|kl⟩ − ⟨i j|l k⟩ is the antisymmetrized variant of the two-electron integral ⟨i j|kl⟩ ≡ ⟨i(1) −1 j(2)|r 12 |k(1)l(2)⟩ involving the spinorbitals with the indices i, j, k, and l. Here and in the following, a, b, and c are the indices of the spinorbitals included in Ψ0, whereas p and q denote those belonging to their orthogonal complement (i.e., the virtual spinorbitals). The respective first-order contributions, J (1)(Q) =
1 ⟨ab|ab⟩ 2 ab
(8)
and A. The weak-correlation limit
At the ω → ∞ (strong-confinement or weak-correlation) limit, the total energy E(Q; ω) of a given electronic state Q of a harmonium atom possesses the asymptotics9,11–13,15,17 E(Q; ω) =
∞
E ( j)(Q) ω(2− j)/2.
(4)
j=0
K (1)(Q) = −
1 ⟨ab|ba⟩, 2 ab
(9)
to the Coulomb and exchange components of W (Q; ω) [J(Q; ω) and K(Q; ω), respectively] are trivial to compute. On the other hand, derivation of the second-order expressions, J (2)(Q) = −2 Re (E p − E a )−1 ⟨ab|pb⟩ ⟨pc∥ac⟩ (10) abc p
At present, closed-form expressions for the coefficients E ( j)(Q) with 0 ≤ j ≤ 2 are known for the 1 S+ and 3 P− electronic states of the two-electron harmonium atom,9,11–13,15,17 the 2 P− and 4 P+ states of the three-electron species,15,17 and the 3 P+, 1 D+, 1 S+, and 5 S− states of the four-electron system.17 In addition, the coefficient E (3)(1 S+) is available in a closed form for the two-electron harmonium atom.12
Jα(2)β (Q) = −Re
and K (2)(Q) = 2 Re
(E p − E a )−1 ⟨ab|bp⟩ ⟨pc∥ac⟩,
(11)
abc p
is somewhat more involved. The expressions for the individual spin-pair components, such as
⟨a β bα |pβ bα ⟩ ⟨pβ c∥a β c⟩ + ⟨aα bβ |pα bβ ⟩ ⟨pα c∥aα c⟩ , E p − Ea abc p
(12)
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TABLE I. The first-order contributions to the components of the electronelectron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom at the weak-correlation limit.a
K (1)(Q)
Q = 2 P− √ (209/120) 2/π √ (1/2) 2/π √ (11/6) 2/π √ (163/40) 2/π √ −(43/40) 2/π √ −(1/2) 2/π √ −(63/40) 2/π
U(1)(Q)
0
Component (1)
Jαα (Q) (1)
J β β (Q) (1) Jα β (Q) J(1)(Q) (1)
K αα (Q) (1)
K β β (Q)
a See
Computation of the sums that enter Eqs. (8)–(13) involves the previously published algebraic approaches15,17 that employ summation techniques based upon properties of hypergeometric functions.35 All the second-order quantities are reducible to the standard form √ 3 1 ln 2 F (C1,C2,C3,C4,C5) = C1 + C2 + C3 + C4 π π π √ ln(1 + 3) C5, (14) + π where the coefficients {C1,C2,C3,C4,C5} are rational numbers.
Q = 4 P+ √ (37/10) 2/π 0 0 √ (37/10) 2/π √ −(17/10) 2/π 0 √ −(17/10) 2/π
B. The strong-correlation limit
0
At the ω → 0 (weak-confinement or strong-correlation) limit, E(Q; ω) possesses the asymptotics9,14,16,20,21
(1)
Eqs. (8) and (9); the entries labeled Jα β (Q) actually list the sums of two equal (1)
(1)
contributions Jα β (Q) and J βα (Q). The spin-summed components are boldfaced.
E(Q; ω) =
2 1 (E p + E q − E a − E b )−1 ⟨ab∥pq⟩ , 2 ab pq
E˜ ( j)(Q) ω(2+ j)/3.
Analogous power series describe the asymptotic behavior of the electron-electron repulsion energy [where a simple scaling argument yields W˜ ( j)(Q) = 32 (1 − j) E˜ ( j)(Q)] and its Coulomb component.18 The exchange component of the electron-electron repulsion energy also possesses the small-ω power expansion
(13)
K(Q; ω) =
∞
K˜ ( j)(Q) ω(5+2 j)/6.
The expansions for the individual spin-pair components [i.e., Jαα (Q; ω), Jα β (Q; ω), Jβα (Q; ω), Jβ β (Q; ω), Kαα (Q; ω), and K β β (Q; ω)] are analogous to those for their spin-summed counterparts.
TABLE II. The second-order contributions to the components of the electron-electron repulsion energies of the 2 P and 4 P states of the three-electron harmonium atom at the weak correlation limit.a − + C1
C2
C3
C4
C5
Numerical value
(2) Jαα (2 P−)
208/27
−6467/270
23/45
−1991/30
1991/45
−0.126 865 317
(2) J β β (2 P−) (2) Jα β (2 P−)
8/3
−8
0
−22
44/3
−0.041 662 624
1216/135
−3731/135
16/45
−226/3
452/9
−0.147 920 981
872/45
−5363/90
13/15
−1637/10
1637/15
−0.316 448 922
J(2)(2P−) (2) K αα (2 P−)
−688/135
815/54
1/9
1231/30
−1231/45
0.070 959 049
K β β (2 P−)
−8/3
8
0
22
−44/3
0.041 662 624
K (2)(2P−)
−1048/135
1247/54
1/9
1891/30
−1891/45
0.112 621 673
U(2)(2P−)
−98/135
967/135
−14/45
43
−310/9
−0.149 481 001
Jαα (4 P+)
592/45
−2096/45
64/15
−592/5
1184/15
−0.189 323 324
(2) J β β (4 P+) (2) 4 J α β ( P +) J(2)(4P+)
0
0
0
0
0
0
0
0
0
0
(2)
(2)
592/45
−2096/45
64/15
−592/5
1184/15
−0.189 323 324
K αα (4 P+)
−368/45
944/45
32/15
272/5
−544/15
0.076 035 997
(2) K β β (4 P+) K (2)(4P+)
0
0
0
0
0
−368/45
944/45
32/15
272/5
−544/15
0.076 035 997
2/15
64/15
−16/15
80/3
−64/3
−0.037 933 367
(2)
U(2)(4P+) a See
Eqs. (10)–(14); the entries labeled spin-summed components are boldfaced.
(2) Jα β (Q)
(16)
j=0
i.e., the second-order contribution U (2)(Q) to the correlation component U(Q; ω) of W (Q; ω) equals twice the second term in the r.h.s. of Eq. (7) or, in other words, twice the second-order contribution to the correlation energy.
Component
(15)
j=0
where the subscripts α and β indicate spin restrictions, are obtained in a similar fashion. The first term in the r.h.s. of Eq. (7), which accounts for the second-order contribution to the Hartree-Fock energy, is readily shown to equal one-half of the sum J (2)(Q) + K (2)(Q). Consequently, U (2)(Q) = −
∞
actually list the sums of two equal contributions
(2) Jα β (Q)
(2)
and J βα (Q). The
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Jerzy Cioslowski
J. Chem. Phys. 142, 114105 (2015)
TABLE III. The leading coefficients of the small-ω expansions for the Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Q = 2 P−
Q = 4 P+
(0) J˜αα (Q)
(17/8) 31/6
(117/25) 31/6
(0) J˜β β (Q)
(13/25) 31/6
0
(0) J˜α β (Q) J˜ (0)(Q)
(19/10) 31/6
0
(909/200) 31/6
(117/25) 31/6
(0) K˜ αα (Q)
−2.335 685 308
−4.826 270 917
(0) K˜ β β (Q)
−0.536 252 324
0
Component
K˜ (0)(Q)
−2.871 937 632
−4.826 270 917
˜ (0)(Q) U
32/3 − (909/200) 31/6
32/3 − (117/25) 31/6
˜ (1)(Q) U
2.871 937 632
4.826 270 917
a See
Eqs. (15)–(17); the entries labeled (0)
(0) J˜α β (Q)
actually list the sums of two equal
(0)
contributions J˜α β (Q) and J˜βα (Q). The spin-summed components are boldfaced.
The small-ω behavior of the correlation component U(Q; ω) is described by yet another series, i.e., U(Q; ω) =
∞
U˜ ( j)(Q) ω(4+ j)/6,
(17)
j=0
with U˜ (0)(Q) = W˜ (0)(Q) − J˜(0)(Q) and U˜ (1)(Q) = −K˜ (0)(Q) [note the mismatch in the powers of ω that enter Eqs. (15) and (16)].
are obtained from Eqs. (2) and (3) by employing finite-basis representations of the pertinent 1-matrices. Computation of the relevant matrix elements involving the basis functions φ I (r) = ζ 3/4 H I x ( ζ x)H I y ( ζ y)H Iz ( ζ z) exp(−ζ r 2/2), (18) where Hk (x) is the kth normalized Hermite polynomial and I ≡ (I x , I y , Iz ), has been described elsewhere.20 In practice, limiting this basis set to the functions with 0 ≤ I x , I y , Iz ≤ 10 is found to produce sufficiently accurate data provided the exponent ζ is optimized de novo for each value of ω. Two-electron integrals involving these functions, which are necessary for evaluations of the r.h.s. of Eqs. (2) and (3), are readily available from the algorithm of McMurchie and Davidson.36 The correlation components are computed simply as the differences between the electron-electron repulsion energies and the sums of their Coulomb and exchange components. The recently introduced robust interpolation between weak- and strong-correlation regimes of quantum systems37 provides a recipe for construction of approximants that accurately reproduce exact values of observables while conforming to their asymptotics at both limits. In the present case, M known approximants for the Coulomb components of the the 0+1 electron-electron repulsion energy are given by J(Q; ω) =
M
CM,k (1 − t) M−k−1 t k+2,
(19)
k=0
where t is the real-valued solution of the equation ω = ω0 (1 − t)−2 t 3
C. Numerical calculations for finite confinement strengths
For finite confinement strengths, the Coulomb and exchange components of the electron-electron repulsion energy
with a parameter ω0. The coefficients CM,0, CM, M , and CM, M−1 are derived from J˜(0)(Q), J (1)(Q), and J (2)(Q) (or their spinpair components), and ω0, which in turn is obtained (together
TABLE IV. The Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the 2 P− ground state of the three-electron harmonium atom.a ω 1000 500 200 100 50 20 10 5 2 1 0.5 0.2 0.1 0.05 0.02 0.01 0.005 0.002 0.001 a The
(20)
Jαα (2 P−;ω)
J β β (2 P−;ω)
Jα β (2 P−;ω)
K αα (2 P−;ω)
K β β (2 P−;ω)
U (2 P−;ω)
43.818 271 30.947 451 19.527 037 13.771 460 9.702 027 6.091 906 4.273 310 2.988 487 1.850 763 1.279 800 0.878 864 0.528 163 0.355 728 0.237 563 0.137 631 0.090 309 0.058 882 0.033 186 0.021 406
12.574 127 8.879 137 5.600 517 3.948 162 2.779 855 1.743 365 1.221 184 0.852 216 0.525 408 0.361 351 0.246 150 0.145 534 0.096 345 0.063 032 0.035 521 0.022 915 0.014 766 0.008 249 0.005 302
46.110 083 32.561 832 20.540 312 14.481 773 10.198 111 6.397 898 4.483 501 3.130 961 1.933 240 1.332 196 0.910 261 0.541 655 0.361 059 0.238 110 0.135 470 0.087 764 0.056 612 0.031 557 0.020 220
−27.052 493 −19.108 066 −12.058 639 −8.505 637 −5.993 165 −3.763 531 −2.639 567 −1.844 572 −1.138 840 −0.783 083 −0.531 831 −0.310 426 −0.201 188 −0.127 025 −0.066 520 −0.039 801 −0.023 459 −0.011 481 −0.006 630
−12.573 700 −8.878 536 −5.599 573 −3.946 837 −2.778 000 −1.740 492 −1.217 217 −0.846 796 −0.517 422 −0.350 950 −0.233 104 −0.129 361 −0.078 915 −0.045 919 −0.021 169 −0.011 569 −0.006 357 −0.002 932 −0.001 647
−0.149 017 −0.148 825 −0.148 444 −0.148 016 −0.147 411 −0.146 217 −0.144 875 −0.142 986 −0.139 254 −0.135 070 −0.129 209 −0.117 926 −0.106 122 −0.091 573 −0.069 922 −0.053 929 −0.039 954 −0.025 655 −0.017 886
entries labeled Jα β (2 P−; ω) actually list the sums of two equal contributions Jα β (2 P−; ω) and J βα (2 P−; ω).
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TABLE V. The Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the 4 P+ first excited state of the three-electron harmonium atom. ω 1000 500 200 100 50 20 10 5 2 1 0.5 0.2 0.1 0.05 0.02 0.01 0.005 0.002 0.001
Jαα (4 P+; ω)
K αα (4 P+; ω)
U(4 P+; ω)
93.167 137 65.824 057 41.561 951 29.334 165 20.688 175 13.017 104 9.151 765 6.419 651 3.997 682 2.779 507 1.921 101 1.165 219 0.789 781 0.529 862 0.307 761 0.201 901 0.131 493 0.073 984 0.047 679
−42.817 185 −30.254 031 −19.106 343 −13.487 913 −9.515 069 −5.989 820 −4.213 085 −2.956 743 −1.842 008 −1.280 323 −0.883 451 −0.532 229 −0.356 530 −0.234 138 −0.129 360 −0.080 045 −0.048 273 −0.023 974 −0.013 916
−0.037 935 −0.037 936 −0.037 937 −0.037 938 −0.037 939 −0.037 937 −0.037 930 −0.037 913 −0.037 851 −0.037 742 −0.037 519 −0.036 885 −0.035 942 −0.034 368 −0.030 989 −0.027 392 −0.023 119 −0.017 168 −0.013 023
with the remaining coefficients {CM,k }) by minimizing the differences between the values of J(Q; ω) computed at 19 confinement strengths ranging from 10−3 to 103 20 and their respective estimates obtained from Eqs. (19) and (20). The M construction of the 0+1 approximants for the exchange components of the electron-electron repulsion energy proceeds analogously, with K˜ (0)(Q), K (1)(Q), and K (2)(Q) serving as the asymptotic constraints, and K(Q; ω) =
M
CM,k (1 − t) M −k−1 t k+5/2
(21)
k=0
replacing Eq. (19). The asymptotic data employed for the correlation compo(2) ˜ (0) ˜ (1) nent M consist of U (Q), U (Q), and U (Q), affording the 1+0 approximants
U(Q; ω) =
M
CM,k (1 − t) M −k t k+4,
where t is the real-valued solution of the equation ω = ω0 (1 − t)−2 t 6.
ω0 C 9,0 C 9,1 C 9,2 C 9,3 C 9,4 C 9,5 C 9,6 C 9,7 C 9,8 C 9,9
III. RESULTS AND DISCUSSION
The first- and second-order contributions to the Coulomb, exchange, and correlation components of the electron-electron repulsion energies of weakly correlated three-electron harmonium atoms in the 2 P− and 4 P+ states are compiled in Tables I and II. As expected, the first-order contributions U (1)(Q) to the correlation components equal zero and their second-order counterparts U (2)(Q) equal twice the asymptotic values of the respective correlation energies that have been obtained previously.15,17 In the doublet ground state, there is a perfect cancellation of the Jβ β (Q; ω) and K β β (Q; ω) components at both orders. At the ω → ∞ limit, the J(Q; ω)/W (Q; ω) ratios 37 for the 2 P− and 4 P+ states tend to 163 100 and 20 , respectively. The corresponding ratios for the exchange components approach 63 − 100 and − 17 20 , whereas those for the correlation components vanish. An entirely different picture emerges at the strong-correlation limit. With their leading asymptotics of K˜ (0)(Q) ω5/6, where the coefficients K˜ (0)(Q) are given by rather complicated expressions,18 the exchange components now become vanishingly small in comparison to their Coulomb and correlation counterparts. Inspection of Table III reveals the large asymp√ √ 39 totic values ( 303 3 ≈ 2.624 and 3 ≈ 2.702 for the 2 P− 200 25 4 and P+ states, respectively) of the J(Q; ω)/W (Q; ω) ratios at ω → 0 that indicate the presence of strong electron correlation effects in three-electron harmonium atoms within the weakconfinement regime.
Jαα (2 P−; ω)
J β β (2 P−; ω)
Jα β (2 P−; ω)
Jαα (4 P+; ω)
0.120 165 477 3 0.621 439 642 4 4.697 842 809 20.719 085 78 44.592 416 70 67.420 189 40 66.262 989 58 43.473 857 06 18.233 810 33 4.449 477 652 0.481 720 312 5
0.013 758 377 00 0.035 856 538 08 0.310 490 678 2 1.167 362 000 3.002 302 178 4.382 920 766 4.434 401 749 3.221 902 289 1.506 790 895 0.402 883 956 1 0.046 794 376 89
0.006 884 684 572 0.082 577 774 64 0.790 857 833 5 2.619 965 718 7.711 306 022 10.330 821 41 11.593 396 26 7.820 528 814 3.640 036 565 1.005 126 325 0.121 373 400 6
0.043 007 883 14 0.689 919 002 4 5.601 792 151 23.195 436 18 53.139 933 18 81.615 693 60 81.642 281 87 54.157 232 15 22.910 796 07 5.626 877 912 0.612 231 709 0
Eqs. (19) and (20); the entries labeled Jα β (2 P−; ω) actually list the parameters for the sum of two equal components Jα β (2 P−; ω) and J βα (2 P−; ω).
a See
(23)
Another, less convenient alternative involves subtracting the sum of the approximants for J(Q; ω) and K(Q; ω) [Eqs. (19)–(21)] from that for the electron-electron repulsion energy (which in turn is readily derived from the published approximant for the total energy20).
9 TABLE VI. The parameters of the 0+1 approximants for the Coulomb components of the electron-electron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Parameter
(22)
k=0
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9 approximants for the exchange components of the electron-electron TABLE VII. The parameters of the 0+1 repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Parameter ω0 C 9,0 C 9,1 C 9,2 C 9,3 C 9,4 C 9,5 C 9,6 C 9,7 C 9,8 C 9,9 a See
K αα (2 P−; ω)
K β β (2 P−; ω)
K αα (4 P+; ω)
0.081 252 946 53 −0.288 367 908 9 −2.669 235 156 −10.969 877 66 −24.633 118 17 −42.150 020 38 −40.308 258 33 −25.595 680 75 −10.232 089 29 −2.373 980 273 −0.244 493 932 1
0.007 125 965 535 −0.008 711 085 680 0.280 317 468 1 −2.208 794 758 2.770 510 883 −5.069 709 999 −0.154 998 307 2 −1.213 721 680 −0.964 374 841 6 −0.295 106 234 0 −0.033 676 885 83
0.015 346 671 11 −0.148 578 801 8 −1.650 428 016 −5.226 123 310 −14.971 359 03 −24.116 716 18 −24.822 273 96 −16.359 252 01 −6.755 137 165 −1.604 299 786 −0.168 033 578 3
Eqs.(20) and (21).
The computed values of the components of the electronelectron repulsion energies at finite confinement strengths are listed in Tables IV and V. As expected, the magnitudes of the correlation components decrease with the state multiplicity. This decrease is more pronounced within the strongconfinement regime which, coupled with the small difference in the respective ω → 0 asymptotic values (see Table III), results in a markedly reduced variation of U(Q; ω) with ω in the case of the 4 P+ state. On the other hand, weakening of the confinement brings about significantly faster decay of the exchange components relative to their Coulomb and correlation counterparts in both states under study. Whereas the aforediscussed energy components allow for assessing the accuracy of approximate electronic structure methods at 19 different values of ω, the approximants (19)–(23) provide the benchmarking data at arbitrary confinement strengths. At M = 9, these approximants faithfully repro 9 duce the data compiled in Tables IV and V. Thus, the 0+1 approximants with the parameters listed in Table VI deviate by at most 2.9, 0.6, 0.8, and 1.2 µhartree, respectively, from 2 the actual values of Jαα (2 P−; ω), Jβ β (2 P−; ω), 9Jα β ( P−; ω), and 4 Jαα ( P+; ω). The respective figures for the 0+1 approximants for Kαα (2 P−; ω), K β β (2 P−; ω), and Kαα (4 P+; ω) (Table VII) 9 are 2.1, 0.9, and 2.5 µhartree, whereas those for the 1+0
approximants for U(2 P−; ω) and U(4 P+; ω) (Table VIII) are 2.6 and 0.5 µhartree.
IV. CONCLUSIONS
The availability of highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of harmonium atoms with a wide range of confinement strengths opens an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are particularly useful in the context of benchmarking of approximate DMFT functionals. The results of such studies will be published elsewhere. The computed data are found to be consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Consequently, robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to those limits are readily constructed, further enhancing the utility of manyelectron harmonium atoms as model systems. ACKNOWLEDGMENTS
9 TABLE VIII. The parameters of the 1+0 approximants for the correlation components of the electron-electron repulsion energies of the 2 P− and 4 P+ states of the three-electron harmonium atom.a Parameter ω0 C 9,0 C 9,1 C 9,2 C 9,3 C 9,4 C 9,5 C 9,6 C 9,7 C 9,8 C 9,9 a See
Eqs. (22) and (23).
U(2 P−; ω)
U(4 P+; ω)
0.002 904 388 173 −0.068 767 811 52 −0.688 517 931 8 −124.378 520 9 247.823 608 9 −215.416 588 9 56.043 826 18 −16.901 055 23 −7.688 967 569 −1.671 063 414 −0.149 481 001 0
0.001 983 615 922 −0.055 891 435 04 −0.550 536 791 8 4.730 618 959 −21.398 513 44 3.960 747 515 −12.792 391 46 −8.588 516 527 −2.846 252 362 −0.494 850 754 8 −0.037 933 366 60
The research described in this publication has been funded by NCN (Poland) under Grant No. DEC-2012/07/B/ST4/ 00553. The support from MPI PKS Dresden is also acknowledged. 1E.
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