J. Phys. Chem. 1992, 96, 627-630
627
Finite Element Multiconfiguration Hartree-Fock Calculations on Carbon, Oxygen, and Neon: The Nuclear Quadrupole Moments of llC, "0, and *lNe Dage Sundbolm* Department of Chemistry, University of Helsinki, Et. Hesperiank. 4, SF-001 00 Helsinki, Finland
and Jeppe Olsen Theoretical Chemistry, Chemical Centre, University of Lund, P.O. Box 124, S-22100 Lund, Sweden (Received: June 17, 1991)
The electric field gradients at the nuclei of C(~P~;~PC), O(~P~;~PC), and Ne(2~~3s';~PO) have been studied as a function of the active space using a newly developed finite-element multiconfiguration Hartree-Fock method. By combination of the calculated electric field gradients with the experimental nuclear quadrupole coupling constants, the values for the nuclear quadrupole moments of "C, 170and , 21Nebecome 0.033 27 (24), -0.025 58 (22), and 0.101 55 (75) b, respectively. The estimated uncertainties are given within parentheses.
1. Introduction The nuclear quadrupole moment (Q), which is one of the few model independent nuclear properties, has traditionally been estimated by comparing measured electric and magnetic hyperfine coupling constants. The quadrupole shielding effects were then accounted for by using Sternheimer corrections.' Alternatively, the calculated electric field gradient from a goodquality ab initio calculation and the experimental nuclear quadrupole coupling constant could be used. The nuclear quadrupole moments can further be determined by measuring the quadrupole splitting in muonic atoms, Le. in atoms where one electron is replaced by a muon.24 For muonic atoms, the electric field gradient can be calculated analytically because the muon moves almost independently of the electrons, and the muon is only slightly influenced by finite nuclear size. Nuclear scattering experiment^^-^ and nuclear theory calculati~ns~*~~ have also been used to obtain values for nuclear quadrupole moments. In this work, the nuclear quadrupole moments of "C, I7O,and ZINeare deduced from large-scale numerical multiconfiguration Hartree-Fock calculations. The relation between the nuclear quadrupole coupling constant (in MHz), the electric field gradient q (in au), and the nuclear quadrupole moment Q (in barn, 1 b = mZ),is
Q = -(eqQ/h)/(234.9647q)
(1.1)
Recently, the nuclear quadrupole moments of the 9Be,141°B,ls (1) Sternheimer, R. M. 2.Naturforsch., A : Phys., Phys. Chem., Kosmophys. 1986, 41A, 24, and references therein. (2) Dey, W.; Ebersold, P.; Leisi, H. J.; Schenk, F.; Walter, H. K.; Zehnder, A. Nucl. Phys. A 1979, 326,418. (3) Weber, R.; Jeckelmann, B.; Kern, J.; Kiebele, U.; Aas, B.; Beer, W.; Beltrami, I.; Bos, K.; de Chambier, G.; Goudsmit, P. F. A.; Leisi, H. J.; Ruckstuhl, W.; Strassner, G.; Vacchi, A. Nucl. Phys. A 1982, 377, 361. (4) Jeckelmann, B.; Beer, W.; Beltrami, I.; de Boer, F. W. N.; de Chambrier, G.; Goudsmit, P. F. A.; Kern, J.; Leisi, H. J.; Ruckstuhl, W.; Vacchi, A. Nucl. Phys. A 1983,408,495. (5) de Boer, J.; Eichler, J. In Advances in Nuclear Physics; Baranger, M., Vogt, E., Eds.; Plenum: New York, 1968; Vol. 1, p 1. (6) Spear,R. H. Phys. Rep. 1981, 73, 369. (7) Vermeer, W. J.; Spear, R. H.; Barker, F. C. Nucl. Phys. A 1989,500, 212. (8) VBlk, H.-G.; Fick, D. Nucl. Phys. A 1991, 530, 475.
(9) Schwalm, D.; Warburton, E. K.; Olness, J. W. Nucl. Phys. A 1977, 293, 425.
(10) Carchidi, M.;Wildenthal, B. H.; Brown, B. A. Phys. Rev. C: Nucl. Phys. 1986, 34, 2280. ( 1 1) Krewald, S.; Schmid, K. W.; Faessler, A. 2.Phys. 1974, 269, 125. (12) van Hees, A. G . M.; Wolters, A. A.; Glaudemans, P. W. M. Nucl. Phys. A 1988,476.69. (13) Brown, B. A.; Wildenthal. B. H. Annu. Rev. Nucl. Part. Sci. 1988. 38, -29, and references therein
0022-3654/92/2096-627$03.00/0
l4N,I625Mg,17and 33S'8isotopes have been determined by combining the electric field gradients ( q or EFG) obtained from finiteelement multiconfiguration HartreeFock calculations with experimental nuclear quadrupole coupling constants. The nuclear quadrupole moments of llBIS and 35S18were deduced using the experimental ratios Q("B)/Q(10B)'9,20and Q(33S)/Q(35S),21 respectively. The nuclear quadrupole moment of the 20.4-min half-life *IC, which is the only carbon nucleus possessing a nuclear quadrupole moment in its ground state, was estimated by Haberstroh et a1.22 to be 30.8 mb from the magnetic and the electric hyperfine coupling constants. First-order configuration calculations of the electric field gradient by Schaefer et al.23yielded a value of 34.26 mb for Q("C). Values of 32.4 and 31.5 mb for Q("C) were obtained from nuclear theory calculations.I2 The value of -25.78 mb for the nuclear quadrupole moment of the very important NMR nucleus 170was estimated by Schaefer et a1.23 from first-order CI calculations on O(3Pe). Many-body perturbation calculationson oxygen by Kelly24yielded a Q(170) of -26.3 mb. The best molecular value for Q(I7O) of -26.4 (3) mb was obtained by Cummins et al.25from coupled pair functional calculations on CO. Nuclear theory calculations yielded values between -21.2 and -31.8 mb for Q(170).10 The quadrupole splittings of noble gas atoms trapped in large molecular systems yield information about the environment of the noble gas atoms. Therefore an accurate value for the nuclear quadrupole moment of 21Nemay become important in NMR spectroscopy.26-28The nuclear quadrupole moment of 93 (10) mb for Z'Ne by Grosof et al.29was estimated by combining the experimental electric and magnetic hyperfine constants. By correcting for quadrupole shielding, Ducas et al.30determined (14) Sundholm, D.; Olsen, J. Chem. Phys. Lett. 1991, 177, 91. (15) Sundholm, D.; Olsen, J. J . Chem. Phys. 1991, 94, 5051. (16) Olsen, J.; Sundholm, D. Submitted for publication in J . Chem. Phys. (17) Sundholm, D.; Olsen, J. NucJ. Phys. A , in press. (18) Sundholm, D.; Olsen, J. Phys. Rev. A 1990, 42, 1160. (19) Dehmelt, H. G. 2.Phys. 1952, 133, 528. (20) Gravina, S. J.; Bray, P. J. J . Magn. Res. 1990, 89, 515. (21) Wentink, T.; Koski, W. S.; Cohen, V. W. Phys. Rev. 1951,81,948. (22) Haberstroh, R. A,; Kossler, W. J.; Ames, 0.;Hamilton, D. R. Phys. Rev. 1964, 136, B932. (23) Schaefer, H. F., 111; Klemm, R. A,; Harris, F. E. Phys. Rev. 1969, 181. . - ., 137. -- .. (24) Kelly, H. P. Phys. Rev. 1969, 180, 55. (25) Cummins, P. L.; Bacskay, G. B.; Hush, N. S . J. Chem. Phys. 1987, 87, 416. (26) Ingman, P.; Jokisaari, J.; Diehl, P. J . Magn. Res. 1991, 92, 163. (27) Diehl, P.; Muenster, 0.; Jokisaari, J. Chem. Phys. Lett. 1991, 178, 147. (28) Ingman, P.; Jokisaari, J.; Pulkkinen, 0.;Diehl, P.; Muenster, 0. Chem. Phys. Lett. 1991, 182, 253. (29) Grosof, G . M.; Buck, P.; Lichten, W.; Rabi, I. I. Phys. Rev. Lett. 1958, 1, 214.
0 1992 American Chemical Society
628 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
Sundholm and Olsen
A multireference all single and double configuration interaction multiconfiguration Hartree-Fock calculation is performed by allowing at most two electrons in the RAS I11 space. In the MR-SDCI-MCHF calculations the orbitals of the RAS I11 space are optimized, while the orbitals of RAS 1 and RAS I1 are the frozen orbitals of a smaller MCHF or a HartreeFock calculation. The following notation is used for the RAS calculations: inactive orbitals//the orbitals of RAS I (minimum number of electrons in RAS I)/the orbitals of RAS II/the orbitals of RAS 2. Methods I11 (maximum number of electrons in RAS 111). When a dash (-) appears in the notation, it means that all the orbitals left of The present work uses a recently developed finite-element multicodguration HartreeFock (FE-MCHF) p r ~ g r a m . ~ " ' * ~ ~ ~ -the ~ ~dash are frozen, while those to the right of the dash are fully energy optimized. The MCHF code is based on the finite-element method and the By use of an LS-coupled wave function with M L = L, the direct configuration-interaction(CI) technique allowing the use electric field gradient at the nucleus is evaluated from of very large CI expansions ( lo6 Slater determinants) together with full orbital optimization in a virtually complete one-electron 4= ~~')IEI%?)) ( ~ ~ 1 4 j ) r (2.2) ij basis. rJ In this numerical method, the radial part of the occupied orIn eq 2.2, riiare the elements of the one-electron density matrix, bitals, the Coulomb and exchange potentials are expanded in I;n are the spherical harmonics, and t # ~is~the radial part of the locally distributed Lagrange interpolation polynomials. In the orbitals. The electric field gradient is spin-independent, i.e. the numerical basis, the energy function which depends on the orbital same q is obtained for all Ms components of the wave function. expansion coefficients (appearing in the integrals) and the conFor systems with an even number of electrons it is, for compufiguration-interaction parameters (appearing in the density matational reasons, most convenient to determine the Ms= 0 comtrices) is constructed ponent of the wave f ~ n c t i o n . ' ~ ~ ' ~ (2.1) E = Chijrij + (1 /2) C gijk/rijk/ When numerical methods are used, the number of basis ij iJ,k,/ functions per symmetry is equal to the number of element functions. In this work, the number of element functions (number where hij and gijkrare the one- and two-electron integrals, reof grid points or nodes) is 201, and fourth-order Lagrange inspectively, and rijand rj,k/ are the elements of the one- and terpolation polynomials are used as local basis functions. The basis two-particle density matrices. The energy (2.1) is optimized with respect to the orbitals and the CI coefficients with imposed orset truncation error in the electric field gradient is less than 10-4 au. thonormality constraints on the occupied orbitals and with a normalized CI vector. The CI coefficients are optimized using 3. Results a Slater-determinant-based direct-CI a l g ~ r i t h m , ~ while ~ . ' ~ the A. Carbon. The valence (1s uncorrelated) spd limit for the orbitals are optimized usihg unconstrained optimization methods electric field gradient (EFG) at the nucleus of 0.645 52 au was of quasi-Newton obtained by systematically increasing the spd space until the EFG We use the restricted active space self-consistent field (RAS became stationary. The ls//2~2pld/2~2p2d-(2)multireference SCF) multiconfiguration Hartree-Fock method34 which is a single and double CI calculation in the ls//4s4p3d CAS orbitals generalization of the complete active space (CAS) SCF method.38 yields the same EFG as the CAS calculation. Triple and quadIn the CAS method, the orbital space is divided into three subruple excitations to orbitals with small occupation numbers do spaces: the inactive, the active, and the virtual (secondary) spaces. not contribute significantly au) to the EFG. The RAS The inactive orbitals are doubly occupied in all configurations, calculations with 252pld reference will produce the same accuracy the virtual orbitals are unoccupied, and full CI is carried out in on the EFG as the corresponding CAS calculation, but with a the set of active orbitals. In the RAS method, the active space significant reduction in the number of configurations. The largest is further subdivided into three active spaces,the RAS I, the RAS valence spd calculation was performed in a ls//2s2pld/ 11, and the RAS I11 spaces. The RAS I space will normally consist 2s2p2d-lslpld(2) set of shells where the frozen shells were the of core and deep valence orbitals, the RAS I1 space consists of ls//4s4p3d ones. valence orbitals, and the orbitals of the RAS I11 space are inThe f-shell contribution to the EFG (An was calculated by troduced to allow for dynamical correlation and polarization. In systematically adding f-shells up to l s / / 2 ~ 2 p l d / 2 ~ 2 ~ 2 d 2 f a RAS calculation a lower limit is given for the number of lslpld2f(2) where the frozen spd shells are the ls//4s4p3d ones electrons in RAS I, and an upper limit is given for the number and the frozen 2f shells are the two f-shells with the largest of electrons in RAS 111, while no restrictions are put on the number occupation numbers in the 1s//2~2pld/2~2p2d-lslpld3f(2) of electrons in RAS 11. calculation. The Afcontribution became 0.001 38 au. Similarly, The magnitude of core-valence correlation effects may be the Ag and the Ah contributions of 0.00024 and O.OOOO8 au, estimated by allowing only single excitations from RAS I to RAS respectively, were obtained. I1 and RAS 111. In these corevalence calculations, the frozen The effect from 1s on the EFG of 4.01474 au was estimated core and valence orbitals are the orbitals of a valence calculation, by performing RAS calculations which allowed single excitations whereas all or some of the orbitals of the RAS I11 space are from 1s. The most important contributions to the EFG came from optimized. The core orbitals have to be frozen, otherwise undesired the s, p, and d orbitals. The core-valence correlation effect can c o r m r e correlation will be introduced by orbital rotations. therefore be estimated in an spd-basis only. In the corevalence calculation, the 1s and the 3s3p2d valence shells of the ls//3s3p2d (30) Dum, T.W.;Feld,M.S.;Ryan, L. W.,Jr.; Skribanowitz, N.; Javan, calculation are frozen, while the corevalence shells of the RAS A. Phys. Rev. A 1972, 5, 1036. I11 space are fully optimized. The relativistic correction is es(31) Olsen, J.; Sundholm, D.To be submitted for publication. timated to be 0.00066 au using the tables of ref 39. (32)Sundholm, D.;Olsen, J. Phys. Rev. A 1990, 42, 2614. (33) Sundholm, D.;Olsen, J. Chem. Phys. Lett. 1990, 171, 53. The final EFG became 0.633 1 (10) au which yields, combined (34)Olsen, J.; Roos, B. 0.;Jsrgensen, P.; Jensen, H. J. A. J . Chem. Phys. with the experimental eqQ/h of -4.949 (28) M H z , ~a ~nuclear 1988.89, 2185. quadrupole moment of 0.03327 (24) b for "C. The carbon (35)Olsen, J.; Jsrgenscn, P.; Simons. J. Chem. Phys. Len. 1990,169,463. calculations are summarized in Tables I and 11. (36)Fletcher, R.Practical Methods of. Oprimizarion; Wiley: New York, . Q(21Ne)more accurately to be 102.9 (75) mb. Nuclear theory calculations on the 21Nenucleus yielded values between 94.7 and 115.8 mb for Q(21Ne)depending on the nuclear model and the Hamilton operator used.lov" Our aim here is to study the electric field gradients at the nuclei of C(2pZ;'PC), O(~P~;~PC), and Ne(2p53s1;'P) as a function of the active space and to provide unambiguous values for the nuclear quadrupole moments of "C, ('0, and 21Ne.
C~GZ(
1980;Vol. 1. (37)Olsen,J.; Yeager, D. L.; Jsrgensen, P. Adv. Chem. Phys. 1-3, 54, 1
(38) Rws, B. 0.Adv. Chem. Phys. 1%7,69, 399,and references therein.
(39)Pyykkb, P.; Pajanne, E.; Inokuti, M. Inr. J . Quantum Chem. 1973, 7, 785.
Nuclear Quadrupole Moments of "C,I7O, and 21Ne
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 629
TABLE I: Total Energy md the Electric Field Gradient of C(3P) as a Function of the Active Space (in 811)
active space"
&/au
9/au
active space"
Edau
q/au
2S//lP ls//2s2pld 1 s// 3s3p2d ls//4s4p3d ls//3s3p2d/lslpld-(2) 1 s//2~2p 1 d/2~2p2d-(2) ls//2~2pld/2~2p2d-l~1 pld(2) 1~//2~2pld/2~2p2d-l~lpldlf(2) 1~//2~2pld/2~2p2d-l~1 pld2f(2) 1 ~//2~2p ld/2~2p2d-lsl p 1 d3f(2) 1~//2~2pld/2~2p2d2f-islpld2f(2) 1~//2~2pld/2~2p2d2f-l~lpldlflg(2) 1~//2~2pld/2~2p2d2f-l~lpldlf2g(2) 1 ~//2~2p ld/2~2p2d2f 1 g-1 s 1 p Id1f1 gl h( 2) IS{ 1 ]/2~2pld/ l~lpld-l~lpld(2) IS( 1 ]/2~2pld/ lslpld-2~2~2d(2) Is(1 I/ lslp/4s4p4d-(2) 1 S{1 ]/ 1 s 1 p/4~4p4d-ls 1 pl d( 2)
-37.688619 -31.114024 -31.780509 -37.781 707 -37.718707 -37.781 703 -37.781 972 -31.186945 -31.181 511 -37.787 639 -37.787 675 -37.788 592 -37.788 782 -31.789080 -37.791 106 -37.192 567 -37.790 944 -37.791 385
0.67672 0.62635 0.64572 0.64552 0.645 52 0.645 52 0.645 52 0.64617 0.64676 0.64683 0.64690 0.64705 0.64707 0.64715 0.63696 0.62989 0.63087 0.631 76
2S//lP ls//292pld ls//3s3p2d 1 ~//2~2p ld/-2~2p2d(2) 1 ~//2~2p ld/2~2p2d-ls 1 pl d( 2) 1 ~//2~2pl d/3~3p3d-l s 1 pl d( 2) IS//l~lp/5~5pSd-l~lpld(2) 1~//2~2pld/5~5pSd-(2) l~//l~lp/6~6psd-ld(2) 1 ~//292pl d/-292p2d 1 f(2) 1~//2~2pld/-2~2~2d2f(2) 1~//2sZpld/3~3~3dlf-(2)~ Is// lslp/4s4p4dlf-(2)b Is//lslp/4s4p4d3f-(2)b ls//lslp/4s4p4d3f-lf(2)b 1 s// 1s 1 p/C4p4d3f-2g(Z)b Is//lslp/4s4p4d3f2g-lg(2) 1 s// 1 s 1 p/3~3p3d2f-ls 1 p Id 1 f1 g 1 h( 2) IS//l~lp/4~4p4d3flg-2h(2) l ~ l]/l~lp/3~3p3d-l~lpld(2) { IS{ 1I/ 1 s 1p/3~3~3d-2~2~2d(2) l~{l)/l~lp/5~5p5d-l~lpld(2)
-74.809398 -14.949 116 -74.968751 -14.972 789 -14.973783 -74.974 107 -74.968204 -74.974230 -74.968 220 -14.990835 -74.993 381 -74.991 617 -74.984945 -14.988 098 -74.988253 -74.992881 -14.993 115 -74.993 254 -74.993 424 -14.983 844 -74.984916 -74.985 315
1.9896 1.7404 1.8098 1.7675 1.7180 1.7637 1.7943 1.7706 1.7926 1.7412 1.7519 1.7579 1.7827 1.7850 1.7863 1.7872 1.1872 1.7878 1.7816 1.7744 1.7581 1.7695
a
For notation, see text.
TABLE Ik Contributions to the Electric FieM Gradient of C ( j P ) (in a d and the C o r r e s m " Nuclear Ourdruwle Moment (in mb)" ~~
TABLE m: Total Jhergy md the Electric Field Gradient of O(3P) as a Function of the Active Space (in au)
~
contribution 9/au Q/mb Hartree-Fock 0.67672 31.12 valence spd limit 0.645 52 32.63 valence spdf limit 0.646 90 32.56 valence spdfg limit 0.647 14 32.55 valence limit 0.64722 32.54 core-valence correction -0.014 14 relativistic correction O.OO0 66 final value 0.633 1 (10) 33.27 (24)b "The nuclear quadrupole coupling constant (eqQ/h) is -4.949 (28) MHz.** bIncludes both the theoretical and the experimental errors.
B. Oxygen. The valence spd limit of 1.7688 au for q is calculated as for carbon by systematically increasing the spd basis. For oxygen, the convergence toward the spd limit is much slower than for carbon. This is probably due to a 2s22p3np1(3P)orbital polarization in the valence shells which is present for oxygen. A similar orbital polarization is not possible for carbon. The frozen shells of the calculations listed in Table I11 were taken from the previous RAS calculation of that table except when otherwise indicated. The largest valence spd calculation was performed in a ls//lslp/6~6p6d-ld(2) set of shells. The contributions to the EFG of -0.0237 au from triple, quadruple, quintuple, and sextuple excitations from the lslp reference were estimated by comparing the results of the ls//lslp/5s5p5d-lslpld(2) RAS calculation with that of the ls//2s2pld/SsSpSd-(2) CIcalculation performed in the same shells. The Afcontribution to the EFG of -0.0166 au is obtained as follows: The contribution from the first f shell is estimated as the difference between the electric field gradients obtained in the 1s//2s2pl d/2s2p2d-l s 1pl d(2) and the 1s//2s2pld/3s3p3d 1f-(2) calculations, respectively. The differential contributions from additional f shells are obtained from the ls//lslp/4~4p4dlf-(2), the 1s// 1s 1p/4s4p4d3f-(2), and the 1s// Is1p/4s4p4d3f-l f( 2) calculations, respectively. The effect of higher excitations of -0.0248 au from the lslp reference is in excellent agreement with the estimate above. The Ag and Ah contributions of 0.0022 and 0.0003 au, respectively, are estimated from the g and h shell calculations in Table 111. The I convergence of the electric field gradient for oxygen is slower than that of carbon and neon. Its orbital polarization shells are rather diffuse and therefore more easily deformed by the correlation and polarization shells with larger I values. The corevalence correlation contribution to the EFG was determined by performing RAS calculations which allowed single excitations from 1s. The 1s core and the 4s4p3d valence shells of the ls//2~2~2d/-2s2~2d(2) valence calculation were frozen and the set of shells was augmented by lslpld and 2s2p2d shells
a For notation, see text. bThe frozen shells are the shells of the hshell calculations.
TABLE Iv: Contributions to the Electric Field Gradient of O c P ) (h au) md the Corresponding Nuclear Quadrupole Moment (in mb)"
contribution qlau Olmb Hartree-Fock 1.9896 -22.33 valence spd limit 1.7688 -25.12 valence spdf limit 1.7522 -25.35 valence spdfg limit 1.7544 -25.32 valence limit 1.7547 -25.32 corevalence correction -0.0217 relativistic correction 0.0034 final value 1.736 (10) -25.58 (22)b "The nuclear quadrupole coupling constant (eqQ/h) is 10.438 (30) MHz." bIncludes both the theoretical and the experimental errors. which were fully energy optimized. In the largest core-valence calculation, the 2s2p2d corevalence shells were frozen and further lslpld shells were added and optimized. The corevalence contribution of -0.0217 au is the difference between the EFG obtained from largest core-valence calculation, corrected for the contribution from higher excitations from the lslp reference, and the ls//2~2pld/-2~2p2d(2)result. For oxygen, the relativistic effect on the EFG of 0.0034 au is estimated by performing a quasi-relativistic ls//2s2pld/ 5sSpSd-(2) CI calculation. A relativistic correction of 0.0041 au would be obtained by using the tables of ref 39. The relativistic corrections of 0.0047 au for the EFG by Judd40 and of 0.0044 au by Kelly24are also close to the present value. In the quasirelativistic CI,the mass-velocity and oneelectron Darwin integrals are added to the one-electron integrals, and a nonrelativistic CI is performed. The relativistic change of the EFG will be introduced via the density matrices. The final EFG of 1.736 (10) au yields, combined with the eqQ/h of 10.438 (30) M H Z , a~ nuclear ~ quadrupole moment of -0.02558 (22) b for I'O. The oxygen calculationsare summarized in Tables I11 and IV. C. Neon. In the valence calculation on Ne, both the 1s and the 2s shells are inactive (doubly occupied in all configurations). The valence spd limit is obtained by systematically increasing the spd basis. In order to reduce the size of the configuration expansion, only single and double excitations are allowed from the lslp reference (the 2p and 3s shells). The contribution from the (40) Judd, B. R. In La structure Hyperfine des Atomes et Molecules; Lefebvre, R.,Moser, C., Eds.; Editions du Centre National de la Recherche Scientique: Paris, 1967; p 3 1 1. (41) Harvey, J. S. M.Proc. R. SOC.London, A 1965, 285, 581.
Sundholm and Olsen
630 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 TABLE V Total Energy and the Electric Field Gradient of Ne(3P) as a Function of tbe Active Space (in nu) active mace'' Rlau qlau 4.9786 -127.992315 2s// lP/lS(l) 4.7705 -128.108 897 2s//2s2pld 4.8061 -128.122305 2s//3s3p2d 4.8157 -128.119 929 2s// Is1 p/2~2p2d-(2) 4.8092 -128.122 772 2s// Is1 p / 2 ~ 2 ~ 2 d - l s l p l d ( 2 ) 4.7616 -1 28.132 305 2s// l s l p / 3 ~ 3 ~ 3 d - l f ( 2 ) 4.7696 -1 28.134 525 2s// lslp/3~3~3d-2f(2) 4.7700 -1 28.138 425 2 ~ / 1/s 1p/ 3~3p3d-2f1g(2) 4.7232 -128.206 062 2s(3)/ 1s 1p/3~3p3d-1s 1p 1d( 2) 4.7107 -128.207 520 2 ~ ( 3 )lslp/3~3p3d-2~2p2d(2) / a
For notation, see text.
TABLE VI: Contributions to the Electric Field Gradient of Ne(") (in au) and the Corresponding Nuclear Quadrupole Moment (in mb)" contribution Hartree-Fcck valence spd limit valence spdf limit valence limit core-valence correction relativistic correction final value
qlau 4.9786 4.7997 4.7601 4.7605 -0.0986 0.0131 4.675 (30)
Q/mb 95.36 98.91 99.74 99.73 101.55 :75y
"The nuclear quadrupole coupling constant ( e q Q / h ) is -1 11.55 (10) M H z . ~ ~Includes both the theoretical and the experimental errors.
higher excitations of -0.0096 au is estimated as the difference between the EFG obtained from the 2s//3s3p2d CAS calculation and the 2s// lslp/2s2p2d-(2) CI calculation. The largest spd calculation yields an EFG of 4.8092 au, and after correction for the higher excitations the valence spd limit becomes 4.7997 au. The Af contribution to the EFG of -0.0396 au and the Ag contribution to the EFG of 0.0004 au are estimated by augmenting the orbital space with orbitals off and g symmetry, respectively. The core-valence contribution to the EFG of -0.0986 au is obtained by allowing in the RAS calculation single excitations from 1s and 2s. In the core-valence calculations, the 2s//2s2p2dIslpld(2) set of shells was augmented by lslpld and 2s2p2d shells. The second set of core-valence shells contributes almost 1 order of magnitude less to the EFG than the first set of corevalence shells. The relativistic correction to the EFG of 0.0131 au is estimated by performing a quasi-relativistic CI in the 2s/ /lslp/3s3p3d-(2) valence shells. The final EFG combined with the experimental eqQ/h of -111.55 (10) M H Z yields ~ ~ a Q(Z'Ne) of 0.10155 (75) b. The neon calculations are summarized in Tables V and VI.
TABLE W.Nuclear Qurdrupole Moments of Compared with Literature Values (in mb) method Q(lIC) ref Q("O) magnetic hfs 30.8 (6) 22 magnetic hfs" nuclear theory 31.5 12 -21.2 UHFb 30.8 42 -25.41 polarization CI 32.17 43 -25.62 first-order C I 34.26 23 -25.78 MBFT -26.3 CI -24 CPFd -26.4 (3) Hartree-FocF 31.12 PW -22.33 numerical M C H F 33.27 PW -25.58 (24) (22)
"C, "0, and 21Ne ref
Q(2'Ne) ref 93 (10) 29 102.9 (75) 30 108.4 11
10 42 43 23 24 44 25 PW 95.36 PW 101.55 (75)
PW PW
a Including a Sternheimer correction'. bunrestricted H a r t r t t F o c k calculations. Many-body perturbation calculation. Coupled pair functional calculation on CO. cPW, present work.
HartreeFock values for Q("C), Q(170), and Q(ZINe)are compared to literature values. For carbon, the previous polari~ation~~ and first-order C123calculations provided electric field gradients with uncertainties of about 3-4%. For oxygen, the accuracy of the CI calculations by the same authors must be about the same as for carbon, and the close agreement with the present best result for Q(170) must be considered fortuitous. The stated error bars of 1S% for the EFG of the many-body perturbation calculation by Kellyz4are about half of the actual uncertainty. Only the experimental uncertainties are included in the error bars of the best molecular value for Q("O) of -26.4 (3) mb?s For neon, there are no previous calculations. The best literature value of 102.9 (75) mb estimated from the electric and magnetic hyperfine constants has error bars of 7.5%, while the present value of 101.55 (75) mb is about 10 times more accurate. As seen from Table VII, electron correlation contributes about 6.5%, 12.5%, and 6.0% to the EFG for C ( ~ P ~ ; ~ POC()~, P ~ ; ~ Pand C ) , Ne(2p53s1;P),respectively.
Acknowledgment. The research reported in this article has been supported by grants from the Academy of Finland, the Swedish Natural Science Research Council (NFR), and the Nordic Council of Ministers. This study has also been supported by IBM Sweden under a joint study contract. The carbon calculations were carried out on the Cray X-MP/432 computer at the Finnish Centre for Scientific Computing and the neon calculations on the IBM-3090 VF computer at the Technical University of Helsinki. The oxygen calculations were run at the Northeast Regional Data Center on the University of Florida IBM 3090VF. Registry No. C, 7440-44-0; 02,7782-44-7; Ne, 7440-01-9; llC, 14333-33-6; I'O, 13968-48-4; 21Ne, 13981-35-6.
4. Discussion
In Table VII, the present Hartree-Fock and multiconfiguration
(43) Schaefer, H. F., 111; Klemm, R. A.; Harris, F. E. Phys. Rev. 1968, 176, 49.
(44) Bessis, N.; Lefebvre-Brion, H.; Moser, C. M. Phys. Rev. 1962, 128, (42) Gcddard, W. A,, 111 Phys. Rev. 1969, 182, 48.
213.
