Properties of harmonium atoms from FCI calculations : Calibration and benchmarks for the ground state of the two-electron species

When used in conjunction with appropriate extrapolation schemes, full configuration interaction (FCI) calculations employing systematic sequences of spherical Gaussian primitives with eventempered exponents shared by functions of different angular momenta are capable of affording ground-state energies of the two-electron harmonium atoms with a few-mHartree accuracy that is sufficient for calibration and benchmarking of approximate electron correlation theories of quantum chemistry. The present approach, which is slated for use in future computations of electronic properties of harmonium atoms with between three and five electrons, calls for a series of 15 FCI runs involving basis sets with between four and eight Gaussian primitives of the sp, spd and spdf type. Its applicability is limited by linear dependencies among basis functions that become significant for small (i.e. less than 0.03) values of the force constant.


Introduction
Exactly (quasi-)solvable models of quantum mechanics that pertain to real physical systems are few and far between.Among nontrivial instances of such models, the two-electron nonrelativistic harmonium atom, [1][2][3] defined by the Hamiltonian with N = 2 (here and in the following the atomic units are used), has attracted much attention from both chemists and physicists.Since harmonium atoms differ from their ordinary (Coulombic) counterparts only in the external potential, they are ideally suited for testing, calibration, and benchmarking of approximate electronic structure methods of quantum chemistry.][6][7][8][9][10][11][12][13] The ground-state energies and wavefunctions of this species are available in closed analytical forms for certain (infinitely many) values of o 2 and as very accurate results of numerical calculations and interpolating approximates otherwise. 3espite being of even greater interest, harmonium atoms with N 4 2 have been studied far less than their two-electron counterparts.With analytical solutions of the respective Schro¨dinger equations unknown, the investigations published so far have either relied on approximate methods or concerned asymptotic limits.Thus, the ground-state properties of the three-electron harmonium have been estimated with the help of an approximate pair model and computed for several values of o with moderate-quality electron correlation methods. 14e ground-state energies and wavefunctions of the three-and four-electron species have been analytically determined at the strong-correlation limit of o -0. 15,16In addition, energies of several states of harmonium atoms have been approximately computed with Monte-Carlo approaches for 2 r N r 8 and a few values of o. 17 The availability of highly accurate electronic structure data for many-electron harmonium atoms would not only provide the means for testing approximate electron-correlation methods on more complicated systems (note that many of these methods become exact in the case of only two electrons being present) but also aid in the reliable theoretical description of three-dimensional quantum dots. 180][21][22] Such crystals emerge in diverse branches of physics including, to name a few, studies of dusty plasmas 23 and ultra-cold ions in electromagnetic traps. 24,25n light of these facts, it would be desirable to develop a robust numerical approach to the computation of electronic properties of harmonium atoms.Such an approach should produce data of benchmark quality for species with diverse (though small) numbers of electrons and magnitudes of the force constant spanning a broad range.Preferably, it should employ only the widely available quantum-chemical software.The full configuration interaction (FCI) method 26 used in conjunction with Gaussian basis sets readily satisfies these criteria.In fact, FCI calculations on two-dimensional quantum dots have been already reported to produce data of useful accuracy. 27he research reported in this paper aims at answering several questions.First of all, it determines the magnitudes of o for which sufficient accuracy can be attained with reasonable computational effort.Second, it addresses the issue of the optimal choices of the basis-set size, the maximum value of the angular momentum, and the exponents of the primitives involved.Third, it tests the viability of extrapolation schemes to the complete-basis-set (CBS) limit.
The present benchmarks open an avenue to calculations on harmonium atoms with between three and five electrons, the results of which will be presented elsewhere.

Numerical procedures
All the calculations described in this paper have employed uncontracted basis sets which, for each value of the angular momentum between 0 and L (0 r L r 3), involve equal numbers N (2 r N r 8) of spherical Gaussian primitives with exponents z k L,N (o) that are even-tempered 28 according to the formula The parameters a L,N (o) and b L,N (o), which depend on N and the maximum angular momentum L but not on the angular momenta of individual primitives, have been optimized (to within the relative accuracy of 10 À6 ) by minimizing the FCI energies E L,N (o) with a combination of simplex and Newton-Raphson methods.
The computed energies E L,N (o) have been extrapolated to the respective N -N limits E L (o) by fitting the actual energy values for N = 4, 5, 6, 7 and 8 with the double-exponential expression which generalizes the Dunning extrapolation. 29The resulting system of five non-linear equations has been solved analytically with the help of the Ramanujan algorithm. 30n turn, the estimates E L (o) have been extrapolated to the respective CBS limits E(o) by fitting the values of E L (o) for L = 1, 2 and 3 with the expression [31][32][33] It should be noted that, although being significantly closer to the exact energies than their unextrapolated counterparts, the estimates E L (o) and E(o) are not variational.All the calculations have been carried out by computing the respective one-and two-electron integrals with the Gaussian03 suite of programs 34 and inputting them into the FCI program of Knowles and Handy. 35

Results
The force constant o measures the magnitude of electron correlation effects in harmonium atoms.In the two-electron species, the critical force constant o crit E 0.04012 constitutes the boundary between the weak-and strong-correlation regimes. 3For o 4 o crit , the system is essentially a set of two three-dimensional harmonic oscillators perturbed by Coulombic coupling.Consequently, it is amenable to description within the perturbation theory, which yields 3 where for the ground-state energy.For o o o crit , the system has the characteristics of the respective Coulomb crystal perturbed by quantum effects due to kinetic energy.The different natures of the two regimes are reflected in the ground-state electron densities r(r), which at r = 0 possess maxima for o 4 o crit and minima for o o o crit , the transition being accompanied by changes in the patterns of the natural orbital occupancies. 3lthough the optimized basis-set parameters a L,N (o) and b L,N (o) vary smoothly with o (Fig. 1 and 2), the calculations become progressively more difficult as o becomes smaller than o crit .This is due to the extent of linear dependences among the primitives with even-tempered exponents increasing rapidly as b L,N (o) -1, which is exactly the behavior observed within the strong-correlation regime.Apparently, reproduction of electron densities that exhibit both minima (for r = 0) and maxima (for some r 4 0) requires combinations of exponential functions with almost degenerate exponents.In practice, FCI calculations with the presently employed Gaussian basis sets are found to be limited to magnitudes of force constant greater than ca.0.03.This conclusion is likely to remain valid for harmonium atoms with more than two electrons.
The dependences of a L,N (o) and b L,N (o) on o follow simple empirical equations.In particular, both the values of Àln(ÀNo 1/2 lna L,N ) (Fig. 1) and Nlnb L,N (Fig. 2) are accurately approximated for N Z 4 by second-order polynomials in o À1/2 , the quadratic contributions diminishing with increasing N.For basis sets composed of two and three primitives, the dependences are less regular, making the corresponding FCI energies E L,2 (o) and E L,3 (o) unsuitable for extrapolation to their N -N limits.Thanks to careful optimization of a L,N (o) and b L,N (o) for each o, these limits are approached quite rapidly (Table 1).For example, the present calculations Fig. 1 The dependence of Àln(ÀNo 1/2 lna 3,N ) on o À1/2 for L = 3.The respective results for L = 0, 1, and 2 are qualitatively identical.The solid lines are fitted second-order polynomials in o À1/2 .yield E 0,8 (1/2) = 2.0343349, which is only 10 À7 higher than that previously obtained with 16 Gaussian functions. 36urther improvements in the accuracy of the FCI energies are afforded by extrapolations, first to the limit of N -N [eqn (3)] and then to the CBS limit [eqn ( 4)].The first extrapolation produces energies E L (o) that comprise partialwave contributions with angular momenta not greater than L to electron correlation (Table 2).Of particular interest are values of E 0 (o), which account for the radial correlation only.These data, computed for 20 different values of o, agree to all digits with the previously published unextrapolated values 36 for o ¼ 5À ffiffiffi ffi 17 p 24 , whereas in four cases (for o = 1/10, 1/2, 10, and 100) they are slightly lower.At the weak correlation limit of o -N, the computed E L (o) conform to eqn (5) with D = À0.0322,À0.0683,À0.0728,and À0.0740 for L = 0, 1, 2, and 3, respectively.
The additional energy lowerings due to the N -N extrapolation are quite significant, reaching 6 Â 10 À6 for large force constants (Fig. 3).It should be noted that attempts to include the values of E L,2 (o) and E L,3 (o) (by augmenting the r.h.s. of eqn (3) with one additional exponential function) produce complex fitting parameters, which in turn yield E L (o) with erratic dependences on o.This behavior can be traced back to the less regular variation of these quantities with o (see above).
Extrapolation to the CBS limit concludes the present approach to the calculation of highly accurate energies of harmonium atoms.][33] The computed values of the parameter C(o) (Table 3) are of the order of unity, confirming the validity of the extrapolation formula.Inspection of the data compiled in Table 4 reveals that the final estimates E(o) are indeed very close to their exact counterparts E exact (o) obtained from highly accurate numerical calculations. 3In general, the relative error in the ground-state energy is found to increase (although not monotonically) with decreasing o.For the three exactly solvable cases (i.e. for o = 1/2, 1/10, and 5À ffiffiffi ffi 17 p 24 ), the absolute errors amount to 2.7 Â 10 À6 , 0.9 Â 10 À6 , and 3.2 Â 10 À6 , respectively.

Discussion and conclusions
When used in conjunction with appropriate extrapolation schemes, full configuration interaction (FCI) calculations employing systematic sequences of spherical Gaussian primitives with even-tempered exponents shared by functions of different Table 1 The optimized basis-set parameters a L,N (o) and b L,N (o) [eqn (2)], and the corresponding FCI energies  ffiffiffi ffi angular momenta are capable of affording ground-state energies of the two-electron harmonium atoms with a few-m Hartree accuracy that is sufficient for calibration and benchmarking of approximate electron correlation theories of quantum chemistry.Although the present prescription, which calls for 15 FCI runs, each involving optimization of the basisset parameters, is not competitive in terms of computational efficiency for the two-electron species, it opens an avenue to systematic, fully automated computations of electronic properties of harmonium atoms with greater numbers of particles.Since in those cases the errors in the energy estimates are expected to be comparable to those presently encountered, this approach is slated to constitute a superior alternative to Monte-Carlo calculations, accuracy of which rapidly degrades with the increasing number of particles. 17urther research is necessary to understand the observed dependences of the optimal basis-set parameters a L,N (o) and b L,N (o) on the force constant o.Being able to predict values of these quantities would significantly decrease numerical effort and thus extend the usefulness of the present approach to even larger harmonium atoms.

Fig. 2
Fig. 2 The dependence of N lnb 3,N on o À1/2 for L = 3.The respective results for L = 0, 1, and 2 are qualitatively identical.The solid lines are fitted second-order polynomials in o À1/2 .

Table 2
The extrapolated FCI energies E L (o)

Table 3
The extrapolated CBS FCI energies E(o), and the fitting parameters B(o) and C(o) [eqn (4)] a a See footnotes on Table2.

Table 4
Comparison of the extrapolated CBS FCI energies E(o) with their exact counterparts E exact (o) a a See footnotes on Table 2.This journal is c the Owner Societies 2010 Phys.Chem.Chem.Phys., 2010, 12, 6712-6716 | 6715 Downloaded by UNIVERSITAT DE GIRONA on 05 August 2010 Published on 27 April 2010 on http://pubs.rsc.org| doi:10.1039/B926389FView Online