Note: The weak-correlation limit of the three-electron harmonium atom

THE JOURNAL OF CHEMICAL PHYSICS 134, 116101 (2011)

Note: The weak-correlation limit of the three-electron harmonium atom Jerzy Cioslowskia) and Eduard Matitob)

Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland

(Received 17 November 2010; accepted 21 January 2011; published online 17 March 2011) [doi:10.1063/1.3553558] Harmonium atoms (HAs), i.e. systems described by the nonrelativistic Hamiltonian 1 Hˆ = − 2

N ! k=1

2 ∇ˆ k +

N !

N

1 ! 2 rkl−1 + ω2 r , 2 k=1 k k>l=1

(1)

are of much interest to both physicists and chemists.1–3 The general forms of the energy asymptotics of these species at the ω → 0 and ω → ∞ limits are well known. At the weakcorrelation limit that corresponds to ω → ∞, the energy E D of the state D is given by the series3–6 E D (ω) =

∞ !

( j) E D ω(2− j)/2

.

(2)

j=0

(1) The zeroth- and first-order energy coefficients, E (0) D and E D , that enter Eq. (2) are trivial to compute. In contrast, evaluation of E (2) D and its high-order counterparts involves infinite summations. At the strong-correlation (or quasiclassical) limit of ω → 0, the energy asymptotics reads1, 3, 7, 8

E D (ω) = E na D (ω) +

∞ !

( j) E˜ D ω(2+ j)/3 ,

(3)

j=0

where the nonanalytical energy term E na D (ω) that depends on the multiplicity of D vanishes at ω = 0. In Eq. (3), E˜ (0) D equals the potential energy of the pertinent spherical Coulomb crystal at its equilibrium geometry.1, 3, 7, 8 The first˜ (2) and second-order energy coefficients, E˜ (1) D and E D , describe the zero-point energy of harmonic vibrations about this equilibrium and the lowest order anharmonic correction to it, respectively.3 The electronic states of HAs are conveniently labeled with the spinorbitals {a} of the three-dimensional harmonic oscillator (with unit mass and force constant) from which the uncorrelated Slater determinant that arises within the zerothorder perturbation theory of the ω → ∞ limit is built. In the widely studied case of the ss singlet ground state of the (0) (1) (2) , E ss , E ss , two-electron HA, closed-form expressions for E ss (0) ˜ (1) (2) 1, 3, 5, 6, 8 ˜ ˜ E ss , E ss , and E ss are presently known. For the analo(0) (1) ˜ (0) , E sp , E spz , gous spz triplet state, only the expressions for E sp z z (1) 6 and E˜ spz have been published. The studies on the three-electron HAs have been quite scarce thus far. The energies of the ss pz doublet ground state and the spx p y quartet first excited state have been computed with Monte Carlo (MC)9 and full configuration-interaction (FCI)10 methods for 3 and 12 values of ω, respectively. Electronic structures of these states have been investigated within 0021-9606/2011/134(11)/116101/2/$30.00

Hartree–Fock and pair-correlation approximations.11–13 In addition, analysis of the strong correlation limit has yielded (0) (1) 5/3 1/2 and E˜ ss + 31/2 ).7 E˜ ss pz = (1/2) 3 pz = (1/2) (3 + 6 In this note, we present closed-form expressions for the zeroth-, first-, and second-order energy coefficients of the weak-correlation limits for the ss pz and spx p y states of the three-electron HA. The key equations that pertain to evaluation of these coefficients read (0) E abc = "a + "b + "c ,

(4)

(1) E abc = %ab| |ab& + %ac| |ac& + %bc| |bc& ,

(5)

and (2) (2) (2) (2) E abc = E ab + E ac + E bc + #a,bc + #b,ac + #c,ab , (6)

where (2) E ab =−

%ab| | pq&2 1!' , 2 pq " p + "q − "a − "b

(7)

and #a,bc = −2

! %ba| |bp& %ca| |cp& ' " p − "a p

! +

'

p

%ac| |bc&2 −%ab| |cb&2 %bc| |ap&2 + , " p + "a − "b − "c " b − "c (8)

the terms with vanishing denominators being excluded from the respective summations. In Eqs. (4), (7) and (8), "a denotes the orbital energy of the spinorbital a. For the ss pz state one obtains (0) E ss pz =

11 2

(9)

and

" 5 2 . (10) = + + = 2 π The second-order energy coefficients for the ss, spz , and s pz states of the two-electron HA turn out to be given by the expressions:

134, 116101-1

(1) E ss pz

(1) E ss

(2) =− E ss

(1) E sp y

E s(1)pz

∞ 2 ! 2−2k (2k − 1)! 2 = 1 − (1 + ln 2) 2 π k=1 (2k + 1) k! π

≈ −0.077891,

(11) © 2011 American Institute of Physics

Downloaded 08 Apr 2011 to 84.88.146.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

116101-2

J. Chem. Phys. 134, 116101 (2011)

J. Cioslowski and E. Matito ∞ 2−2k (2k − 1)! 8 ! 3π k=1 (2k + 1)(2k + 3) k!2

(2) E sp =− z

8 5 (4 + 3 ln 2) ≈ −0.017821, = − 9 27π

where the pertinent intermediate quantities read: (2) (2) (2) = E sp = E (2) E sp px p y = E spz x y

(12)

and E s(2)pz = − =

∞ 1 ! 2−2k (6k + 13)(2k − 1)! 3π k=1 (2k + 1)(2k + 3) k!2

1 7 (43 + 39 ln 2) ≈ −0.047856, − 9 27π

(13)

∞

1 ! 2−4k (−4k 2 +28k +47)(2k − 1)! #s,s pz = − 6π k=1 (2k +1)(2k +3) k!2

√ √ 1 26 − [146+6 3+282 ln 2− 141 ln(2+ 3)] 27 54π ≈ −0.016540, (14)

=

#s,spz

∞ 2 ! 2−4k (2k − 5)(2k − 1)! = 3π k=1 (2k + 1) k!2

√ 2 4 − [6 + 10 ln 2 − 5 ln (2 + 3)] 3 3π ≈ −0.013475,

= and # pz ,ss = −

(15)

∞ 4 ! 2−4k (−2k 2 − k + 5)(2k − 1)! 3π k=1 (2k + 1)(2k + 3) k!2

these coefficients afford the final result ∞ 2−4k (2k − 1)! 1 ! (2) E ss, = − pz 2π k=1 (2k + 1)(2k + 3) k!2 # $ × (−12k 2 + 12k + 49) + 22k (12k + 26)

√ √ 49 1 = + [−88 + 2 3 − 173 ln 2 + 98 ln(1 + 3)] 9 6π ≈ −0.176654. (17)

Analogous calculations for the spx p y state produce: 13 , 2

(1) E sp x py

(1) E sp x

=

(18) +

(1) E sp y

+

E (1) px p y

=2

"

2 , π

(19)

and (2) E sp x py

∞ 8 ! 2−4k (2 + 22k )(2k − 1)! =− π k=1 (2k + 1)(2k + 3) k!2

√ √ 23 8 + [−4 + 3 − 7 ln 2 + 4 ln (1 + 3)] 9 3π ≈ −0.0756103, (20)

=

=

∞ 16 ! 2−4k (2k + 5)(2k − 1)! 15π k=1 (2k + 1)(2k + 3) k!2

√ 16 [−17 + 3 3 + 4π − 30 ln 2 135π √ +15 ln (2 + 3)] ≈ −0.010470,

(22)

and # px ,sp y = # p y ,spx =

∞ 16 ! 2−4k (k − 5)(2k − 1)! = 15π k=1 (2k + 1)(2k + 3) k!2

√ 4 [−86 + 39 3 + 7π − 120 ln 2 135π √ +60 ln (2 + 3)] ≈ −0.005839.

(23)

The newly obtained expressions for the second-order energy coefficients [Eqs. (17) and (20)] resemble those for the twoelectron HA [Eqs. (11)–(13)] but are somewhat more involved. Interestingly, the zeroth-, first-, and second-order energy coefficients of the series (2) and (3) uniquely determine the 13 parameters of the [7/5] Padé approximant F[7/5] (x) that enters the approximate expression E D (ω) ≈ ω2/3 F[7/5] (ω1/6 ),

√ √ 2 22 − [43 − 6 3 + 60 ln 2 − 30 ln(2 + 3)] = 27 27π ≈ −0.003072 (16)

(0) E sp = x py

[which follows from the identical intracular components of the states in question; compare Eq. (14)], #s, px p y = −

(2) (2) the formulas for E ss and E sp matching those obtained z 4–6 previously. When combined with the nonadditivity terms,

(21)

(24)

which conforms to both the small- and large-ω asymptotics.14 In summary, when combined with the previously known small-ω asymptotics, the closed-form second-order expressions for the energies of three-electron harmonium atoms at the weak-correlation limits allow for quite accurate predictions of energies of these species for all magnitudes of the confinement strength. a) Electronic

mail: [email protected]. mail: [email protected]. 1 M. Taut, Phys. Rev. A 48, 3561 (1993). 2 M. Taut, J. Phys. A 27, 1045 (1994). 3 J. Cioslowski and K. Pernal, J. Chem. Phys. 113, 8434 (2000), and the references cited therein. 4 R. J. White and W. Byers-Brown, J. Chem. Phys. 53, 3869 (1970). 5 J. M. Benson and W. Byers-Brown, J. Chem. Phys. 53, 3880 (1970). 6 P. M. W. Gill and D. P. O’Neill, J. Chem. Phys. 122, 094110 (2005). 7 J. Cioslowski and K. Pernal, J. Chem. Phys. 125, 064106 (2006). 8 J. Cioslowski and E. Grzebielucha, Phys. Rev. A 77, 032508 (2008). 9 K. Varga, P. Navratil, J. Usukura, and Y. Suzuki, Phys. Rev. B 63, 205308 (2001). 10 J. Cioslowski and E. Matito, J. Chem. Theory Comput. 11 T. Vorrath and R. Blümel, Eur. Phys. J. B 32, 227 (2003). 12 P. A. Sundqvist, S. Y. Volkov, Y. E. Lozovik, and M. Willander, Phys. Rev. B 66, 075335 (2002). 13 M. Taut, K. Pernal, J. Cioslowski, and V. Staemmler, J. Chem. Phys. 118, 4818 (2003). 14 See supplementary material at http://dx.doi.org/10.1063/1.3553558 for the numerical coefficients that enter Eq. (24) and the accuracy of its energy predictions. b) Electronic

Downloaded 08 Apr 2011 to 84.88.146.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions