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Nuclear motion dependence of the electric field gradient at the 9Be nucleus in BeH+
Volume 196, number 5
CHEMICAL PHYSICS LETTERS
2 1 August 1992
Nuclear motion dependence of the electric field gradient at the 9Be nucleus in BeH+ Antonio C. Borin a, Francisco B.C. Machado b and Fernando R. Ornellas a a Instituto de Quimica, Universidade de .%o Paula. Caixa Postal 20780, Scio Paulo, SP 01498, Brazil b Instituto de Estudos Avancados, Divisdo de Fisica Tedrica, Centro Tkcnico Aeroespacial. Caixa Postal 6044, Srio Josh dos Campos, SP 12231, Brazil
Received 26 May 1992; in final form 1 June 1992
The electric field gradient (q) at the beryllium nucleus in BeH+ has been calculated as a function of the internuclear distance using the best BeH+ wavefunction available in the literature. Nuclear motion effects on q are computed as an average of q(R) over the vibrational wavefunction. It is shown that Buckingham’s expression is inappropriate to compute such a correction and therefore the results reported by Diercksen and Sadlej on BeH+ and other systems do not have the accuracy they imply to have.
1. Introduction
Among the several properties of beryllium isotopes, the nuclear quadrupole moment (Q) has received considerable attention [ l-81. Despite the use of different techniques and methods to establish a definite reliable value, the data reported to date do not seem to show a good agreement. From the molecular point of view, the nuclear quadrupole moment can be determined by combining spectroscopic data (quadrupole coupling constants, eqQ/h, in MHz) and values of the electric field gradient (q, in au) calculated by any of the different post-Hartree-Fock electronic structure methods of quantum chemistry by the expression [ 9 ] Q=-
eqQ/h 2349647q
’
(1)
where, for a linear molecule, q is the parallel (zz) component of the electric field gradient at the nucleus of interest. Eq. ( 1) can be used either with an atomic or molecular calculation of the electric field gradient, but to guarantee accurate and reliable values of q, it is Correspondence to: F.R. Omellas, Instituto de Quimica, Universidade de Slo Paulo, Caixa Postal 20780, SBo Paulo, SP 01498, Brazil.
mandatory to use extended sets of basis functions and a method which includes correlation effects at a high level. The influence of these factors on the value of q has been the subject of several studies [ 10,111. If one is dealing with molecular calculations, the contribution of rotational and vibrational effects on the equilibrium distance value of the electric field gradient must reliably be established. Early in the sixties, Buckingham [ 121 derived an approximate expression incorporating the nuclear motion; its use is clearly dependent on the accuracy of the different spectroscopic constants involved and on the accuracy of the numerical derivatives. More recently, an alternative approach to incorporate nuclear motion in the determination of the nuclear quadrupole moment has also been described by Vojtik et al. [ 13 1. In the case of beryllium, the problem of the molecular determination of Q has been attacked by Diercksen and Sadlej [ 21 using highly correlated electronic approaches. Despite the authors’ emphasis on the importance of both basis sets and correlation effects and their claim on the accuracy of their results, recent work on BeH+ [ 141 has shown that their best wavefunction is still incomplete in describing correlation effects. Besides, the use of an approximate expression to take into account the vibrational and rotational contributions and the use of spectroscopic constants with different accuracies
0009-2614/92/S 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
417
Volume 196, number
5
CHEMICAL
PHYSICS
raises some doubts on the final accuracy of the reported values. A study of the nuclear quadrupole moment of 9Be has also recently been carried out by Sundholm and Olsen [ 31 on the beryllium atom using a numerical multiconfiguration Hartree-Fock approach and by Diercksen et al. [ 15 ] on the Be0 molecule. Further electric properties of Be have been recently reported by Pluta and Kurtz [ 16 1. Using the best wavefunction reported to date on the system BeH+ [ 14 1, the aim of this study was ( 1) to establish a definite and accurate value of the electric field gradient and its dependence on the internuclear distance and (2) to check the validity of Buckingham’s expression to incorporate the nuclear motion contribution to q, thus providing reliable data for the spectroscopic determination of the nuclear quadrupole moment of 9Be.
LETTERS
21 August
$[3(l+ ~)(~)~=o+(g-J
qv=fL+
x(u+1) >
(2)
where B, o, and (Y,are the usual spectroscopic constants, qe is the value of the electric field gradient at the equilibrium distance (R,=2.4797 au) and r= (R-R,) /R,. As mentioned previously, the use of this formula requires accurate values of the spectroscopic constants and of the derivatives of q. The alternative approach of Vojtik et al. [ 131 consists essentially of first computing the electric field gradient with a highly correlated wavefunction over a relatively wide range of internuclear distances, thus obtaining q as a function of R, q(R). Then the effect of nuclear motion is computed as the average of q(R) over the vibrational wavefunction, qv,v=
2. Methodology 2. I. Electronic wavefunction and properties
The methodology employed to calculate the electronic properties is that of the multi-reference single and double excitations configuration interaction (MRSDCI) approach as implemented in the MELD codes [ 17 ] and described in detail elsewhere [ 141. It is, however, worth pointing out here that the molecular orbitals were generated by an internally consistent self-consistent field (ICSCF) [ 18,19 ] calculation and that the atomic basis sets, ( 13s9p4d If) / [ 9s7p4dlf] on beryllium and (8s2p2d)/ [ %2p2d] on hydrogen, comprised a total of 87 contracted functions. Also an energy threshold of 1.Ox 1 O-’ au was used in the selection of the configuration state functions by perturbation theory, thus limiting the size of the final wavefunction to about 10000 terms. CZvpoint group symmetry was used throughout the calculation.
1992
(3)
where v and N stand for the vibrational and rotational quantum numbers. The vibrational wavefunction XuNis a solution of the one-dimensional radial Schrijdinger equation -G
fi= d=X&R) dR=
+
E(R)+
!!iNcN+l) 2~
R=
(4)
where ,u is the reduced mass and E(R) the potential energy function. The functions q(R)and E(R) were generated numerically using a cubic spline fitting to a set of about 20-30 computed points. The vibrational wavefunction was obtained by numerical integration of the radial equation using the NumerovCooley method as implemented in the INTENSITY program [ 201 and then used to vibrationally average q(R).
3. Results and discussion
2.2. Nuclear motion effects in q The effect of nuclear motion on the electric field gradient has usually been computed with an approximate expression known as the Buckingham formula
[121, 418
First, to assess the quality of the one-particle basis set used in this work an SCF calculation of the total energy, dipole moment and electric field gradient was carried out at the equilibrium internuclear distance. These results are summarized in table 1 together with
Volume
196, number
Table 1 Numerical
CHEMICAL
5
values of energies
(E, in au), dipole moments
PHYSICS
Ref. [21] numerical
E 1 q(Be) a) Calculated
MRSDCI
-14.854015 - 1.249639 -0.11541
- 14.937139 - 1.168998 -0.10605
(q, in au) of BeH+ at R=2.4797
(p, in au) and electric field gradients
This work SCF
21 August
LETTERS
1992
au
Ref. [2] HF
- 14.854177 - 1.249255 -0.11551
Ia
SCF
CCSDT-
- 14.853382 - 1.247925 -0.11627
-14.931182 - 1.2151 a) -0.10640
with the CASSCF wavefunction.
Table 2 Variation of q(Be) with the perturbation old for BeH+ at R~2.4797 au. E(SCF)=
energy selection thresh- 14.854015 au
Threshold
No. of CSF
AEcr
q(Be)
1.00x 10-e 0.15x10-6 0.10x 10-e 0.00
5818 9207 9674 68114
-0.082822 -0.083112 -0.083124 -0.083221
-0.10542 -0.10602 -0.10605 -0.10609
the numerical Hartree-Fock values of PyykkS et al. [ 2 1 ] and the calculation of Diercksen and Sadlej [ 21. The data collected in table 1 show clearly that the overall results of this study are superior to those of ref. [ 21; their error in the total energy amounts to 795 uhartree whereas that of the present work is only 162 uhartree; concerning the dipole moment and the electric field gradient at the beryllium nucleus, their errors are 1330 and 760 pau, respectively, whereas Table 3 Electric field gradient
(in au) at the beryllium
and hydrogen
uncertainties about five times smaller, 300 and 100 uau, result in the present work. Concerning now the amount of correlation energy recovered by the final wavefunction, the energy computed at the experimental equilibrium distance in ref. [2], - 14.931182 au, is quite high if compared with our result of - 14.937139 au. The error in the energy, 3.7 kcal/mol relative to our result, is about four times larger than what has been assumed by the theoretical community as a reference value implying chemical accuracy in the calculation. Next was examined the sensitivity of q(Be) to the energy threshold used in the perturbation energy selection of the configuration state functions. The results summarized in table 2 show that q(Be) has practically converged for a threshold in the energy of 1.0x lo-’ hartree. It also shows that the contribution of electron correlation to q(Be) is positive and amounts to 0.00932 au ( z 8Oh).
nuclei in BeH+ as a function
of the internuclear
distance
R
q(Be)
q(H)
R
&Be)
q(H)
1.6 1.7 1.8 2.0 2.1 2.2 2.3 2.4 2.4797 2.5 2.6 2.8 3.0 3.2 3.5
-0.10903 -0.12533 -0.13303 -0.13381 -0.12996 -0.12470 -0.11850 -0.11178 -0.10605 -0.10460 -0.09741 -0.08329 -0.06982 -0.05758 -0.04184
0.98230 0.77982 0.62060 0.39467 0.31493 0.25124 0.2003 1 0.15950 0.13285 0.12672 0.10026 0.06167 0.03640 0.01995 0.00582
3.75 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0
-0.03181 -0.02364 -0.01287 - 0.00708 -0.00388 -0.00213 -0.00058 - 0.00023 - 0.00004 0.00007 0.00000 -0.00001 - 0.00000 - 0.00002
0.00017 -0.00221 -0.00214 -0.00013 0.00107 0.00197 0.00257 0.00220 0.00172 0.00130 0.0004 1 0.00017 0.00009 0.00006
419
Volume 196, number 5
CHEMICAL PHYSICS LETTERS
21 August 1992
BeH+
BeH+
_
0.40
3
O.OO 3 &.J==--
-0.20
1 -0.40
m, 0.00
2.00
4.00
6.00
8.00
R(o.u.) Fig. 1. Electric field gradient at the beryllium and hydrogen nuclei in BeH+ as a function of the internuclear distance.
Fig. 2. Vibrationally averaged electric field gradient in BeH+ as a function of the vibrational quantum number.
The magnitude of q( Be) at the equilibrium internuclear distance computed in this work, -0.10605 au, is a little bit lower than that of ref. [ 2 1,- 0.10640 au, but consistent with their finding that q becomes less negative as the amount of correlation recovered increases. Note also that at the SCF level, the data of ref. [ 21 show that basis set incompleteness in the one-particle space makes q more negative (- 0.11627 au) than the exact numerical Hartree-Fock value of -0.11551 au. Values of the electric field gradient at both the beryllium and hydrogen nuclei as a function of the internuclear distance are collected in table 3 and the
global behavior of these functions is displayed in fig. 1. For hydrogen, q(H) decrease steadily with increasing distance going to zero in the asymptotic limit; a slight minimum is encountered at R z 4.5 au. For beryllium, q(Be) reaches its lowest value at R m 2.0 au and then goes to zero at larger distances. This behavior is very similar to that found by Vojtik et al. [ 131 for the isoelectronic LiH. Vibrationally averaged results of the electric field gradient for selected rotational levels are presented in table 3 together with those computed with the aid of expression (2) of Diercksen and Sadlej [ 21. It is striking to see that the results of ref. [ 21 fail to show
Table 4 Vibrational dependence of the electric field gradient at the Be nucleus computed as a vibrational average and with the aid of expression (2) ofref. [2] V
420
This work
Ref. [2]
N=O
N=l
N=2
N=5
-0.10339 -0.09186 -0.09240 -0.08709 -0.08166 -0.07632 -0.07108 -0.06596 - 0.06093
-0.10336 -0.09782 -0.09237 - 0.08706 -0.08163 -0.07624 -0.07105 -0.06593 - 0.06086
-0.10329 -0.09777 -0.09230 -0.08699 -0.08156 -0.07622 -0.07098 -0.06587 -0.06084
-0.10289 -0.09736 -0.09191 -0.08660 -0.08117 -0.07583 - 0.07060 -0.06549 - 0.06048
-0.10613 -0.10559 -0.10505 -0.10451 -0.10397 -0.10343 -0.10289 -0.10235 -0.10181
Volume 196, number 5
CHEMICAL PHYSICS LETTERS
the strong dependence of q(Be) on the vibrational levels (see also fig. 2 ). In fact, the vibrationally averaged values of q1 and q2 computed in this work already reflect changes of the order of magnitude of the electron correlation corrections. These data clearly show that vibrational corrections introduced via eq. (2) completely spoiled the accuracy implied in the calculation of q. Therefore, the use of that expression and similar ones obtained for other systems do not have the accuracy they imply to have, for the effect of vibrational motion is not properly taken into account. As to the effect of rotation on the value of q, table 4 shows that it is almost negligible.
4. Conclusions This work, by reporting very accurate values of the electric field gradient as a function of the internuclear distance and therefrom computing its vibrational average, shows that Buckingham’s expression is inappropriate for such a correction and that existing calculations on BeH+ and other systems where that formula has been used do not have the accuracy they imply to have.
Acknowledgement The authors are grateful to Professor E.R. Davidson for use of the MELD codes and to Professor W.C. Stwalley for making available a copy of the INTENSITY program. The services of the University of Sgo Paulo Computing Center and of the Data Processing Center of the Instituto de Estudos Avancados are also greatly appreciated. The authors also thank CNPq of Brazil for a research fellowship.
21 August 1992
References [ 1] T. Mertelmeier and H.M. Hofmann, Nucl. Phys. A 459 (1986) 387.
[ 21 G.H.G. Diercksen and A.J. Sadlej, Chem. Phys. Letters 155 (1989) 127.
[ 31 D. Sundholm and J. Olsen, Chem. Phys. Letters 177 ( 1991) 91.
[ 41 A.G. Blachman and A. Laurio, Phys. Rev. 153 ( 1967) 164. [ 51 P. Haghavan, At. Data Nucl. Data Tables 42 ( 1989) 189. [6] S.N. Ray,T.IxeandT.P.Das,Phys.Rev.A8 (1973) 1748. [7] G.H. Fuller, J. Phys. Chem. Ref. Data 5 (1976) 835. [S] CM. Lederer and V.S. Shirley, eds., Table of isotopes (Wiley, New York, 1978) appendices, p. 44. [ 91 E.A.C. Lucken, Nuclear quadrupole coupling constants (Academic Press, New York, 1969 ) . [lo] M. Urban, I. CemuStlk, V. Kellij and J. Noga, in: Methods in computational chemistry, ed. S. Wilson (Plenum Press, New York, 1987) p. 117, and references therein. [ 111 S. Kama and F. Grein, Mol. Phys. 69 ( 1990) 661, [ 121 A.D. Buckingham, J. Chem. Phys. 36 (1962) 3096. [ 131 J. Vojtfk, I. Paidarova, V. Spirko, J. Savrda and M. PetmS, Chem. Phys. Letters 157 (1989) 337; Intern. J. Quantum Chem. 38 (1990) 357; I. Paidarova, J. Vojtfk, L. CeSpiva, J. Savrda and V. Spirko, Intern, J. Quantum Chem. 38 (1990) 283. [ 141 F.B.C. Machado and F.R. Omellas, J. Chem. Phys. 94 (1991) 7237. [ 151 G.H.F. Diercksen, A.J. Sadlej and M. Urban, Chem. Phys. 158 (1991) 19. [ 161 H.A. Pluta and T. Kurtz, Chem. Phys. Letters 189 (1992) 255. [ 171 E.R. Davidson, MELD: a many electron description, in: Modem techniques in computational chemistry: MOTEC91, ed. E. Clementi (ESCOM, Leiden, 1990), ch. 9. [ 181 E.R. Davidson, J. Chem. Phys. 57 (1972) 1999. [ 191 L. Stenkamp and E.R. Davidson, Theoret. Chim. Acta 30 (1973) 283. [ 201 W.T. Zemke and W.C. Stwalley, Program Intensity, QCPE 4 (1981) 79. [ 2 1 ] P. Pyykko, D. Sundholm and L. Laaksonen, Mol. Phys. 60 (1987) 597.
421
CHEMICAL PHYSICS LETTERS
2 1 August 1992
Nuclear motion dependence of the electric field gradient at the 9Be nucleus in BeH+ Antonio C. Borin a, Francisco B.C. Machado b and Fernando R. Ornellas a a Instituto de Quimica, Universidade de .%o Paula. Caixa Postal 20780, Scio Paulo, SP 01498, Brazil b Instituto de Estudos Avancados, Divisdo de Fisica Tedrica, Centro Tkcnico Aeroespacial. Caixa Postal 6044, Srio Josh dos Campos, SP 12231, Brazil
Received 26 May 1992; in final form 1 June 1992
The electric field gradient (q) at the beryllium nucleus in BeH+ has been calculated as a function of the internuclear distance using the best BeH+ wavefunction available in the literature. Nuclear motion effects on q are computed as an average of q(R) over the vibrational wavefunction. It is shown that Buckingham’s expression is inappropriate to compute such a correction and therefore the results reported by Diercksen and Sadlej on BeH+ and other systems do not have the accuracy they imply to have.
1. Introduction
Among the several properties of beryllium isotopes, the nuclear quadrupole moment (Q) has received considerable attention [ l-81. Despite the use of different techniques and methods to establish a definite reliable value, the data reported to date do not seem to show a good agreement. From the molecular point of view, the nuclear quadrupole moment can be determined by combining spectroscopic data (quadrupole coupling constants, eqQ/h, in MHz) and values of the electric field gradient (q, in au) calculated by any of the different post-Hartree-Fock electronic structure methods of quantum chemistry by the expression [ 9 ] Q=-
eqQ/h 2349647q
’
(1)
where, for a linear molecule, q is the parallel (zz) component of the electric field gradient at the nucleus of interest. Eq. ( 1) can be used either with an atomic or molecular calculation of the electric field gradient, but to guarantee accurate and reliable values of q, it is Correspondence to: F.R. Omellas, Instituto de Quimica, Universidade de Slo Paulo, Caixa Postal 20780, SBo Paulo, SP 01498, Brazil.
mandatory to use extended sets of basis functions and a method which includes correlation effects at a high level. The influence of these factors on the value of q has been the subject of several studies [ 10,111. If one is dealing with molecular calculations, the contribution of rotational and vibrational effects on the equilibrium distance value of the electric field gradient must reliably be established. Early in the sixties, Buckingham [ 121 derived an approximate expression incorporating the nuclear motion; its use is clearly dependent on the accuracy of the different spectroscopic constants involved and on the accuracy of the numerical derivatives. More recently, an alternative approach to incorporate nuclear motion in the determination of the nuclear quadrupole moment has also been described by Vojtik et al. [ 13 1. In the case of beryllium, the problem of the molecular determination of Q has been attacked by Diercksen and Sadlej [ 21 using highly correlated electronic approaches. Despite the authors’ emphasis on the importance of both basis sets and correlation effects and their claim on the accuracy of their results, recent work on BeH+ [ 141 has shown that their best wavefunction is still incomplete in describing correlation effects. Besides, the use of an approximate expression to take into account the vibrational and rotational contributions and the use of spectroscopic constants with different accuracies
0009-2614/92/S 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
417
Volume 196, number
5
CHEMICAL
PHYSICS
raises some doubts on the final accuracy of the reported values. A study of the nuclear quadrupole moment of 9Be has also recently been carried out by Sundholm and Olsen [ 31 on the beryllium atom using a numerical multiconfiguration Hartree-Fock approach and by Diercksen et al. [ 15 ] on the Be0 molecule. Further electric properties of Be have been recently reported by Pluta and Kurtz [ 16 1. Using the best wavefunction reported to date on the system BeH+ [ 14 1, the aim of this study was ( 1) to establish a definite and accurate value of the electric field gradient and its dependence on the internuclear distance and (2) to check the validity of Buckingham’s expression to incorporate the nuclear motion contribution to q, thus providing reliable data for the spectroscopic determination of the nuclear quadrupole moment of 9Be.
LETTERS
21 August
$[3(l+ ~)(~)~=o+(g-J
qv=fL+
x(u+1) >
(2)
where B, o, and (Y,are the usual spectroscopic constants, qe is the value of the electric field gradient at the equilibrium distance (R,=2.4797 au) and r= (R-R,) /R,. As mentioned previously, the use of this formula requires accurate values of the spectroscopic constants and of the derivatives of q. The alternative approach of Vojtik et al. [ 131 consists essentially of first computing the electric field gradient with a highly correlated wavefunction over a relatively wide range of internuclear distances, thus obtaining q as a function of R, q(R). Then the effect of nuclear motion is computed as the average of q(R) over the vibrational wavefunction, qv,v=
2. Methodology 2. I. Electronic wavefunction and properties
The methodology employed to calculate the electronic properties is that of the multi-reference single and double excitations configuration interaction (MRSDCI) approach as implemented in the MELD codes [ 17 ] and described in detail elsewhere [ 141. It is, however, worth pointing out here that the molecular orbitals were generated by an internally consistent self-consistent field (ICSCF) [ 18,19 ] calculation and that the atomic basis sets, ( 13s9p4d If) / [ 9s7p4dlf] on beryllium and (8s2p2d)/ [ %2p2d] on hydrogen, comprised a total of 87 contracted functions. Also an energy threshold of 1.Ox 1 O-’ au was used in the selection of the configuration state functions by perturbation theory, thus limiting the size of the final wavefunction to about 10000 terms. CZvpoint group symmetry was used throughout the calculation.
1992
(3)
where v and N stand for the vibrational and rotational quantum numbers. The vibrational wavefunction XuNis a solution of the one-dimensional radial Schrijdinger equation -G
fi= d=X&R) dR=
+
E(R)+
!!iNcN+l) 2~
R=
(4)
where ,u is the reduced mass and E(R) the potential energy function. The functions q(R)and E(R) were generated numerically using a cubic spline fitting to a set of about 20-30 computed points. The vibrational wavefunction was obtained by numerical integration of the radial equation using the NumerovCooley method as implemented in the INTENSITY program [ 201 and then used to vibrationally average q(R).
3. Results and discussion
2.2. Nuclear motion effects in q The effect of nuclear motion on the electric field gradient has usually been computed with an approximate expression known as the Buckingham formula
[121, 418
First, to assess the quality of the one-particle basis set used in this work an SCF calculation of the total energy, dipole moment and electric field gradient was carried out at the equilibrium internuclear distance. These results are summarized in table 1 together with
Volume
196, number
Table 1 Numerical
CHEMICAL
5
values of energies
(E, in au), dipole moments
PHYSICS
Ref. [21] numerical
E 1 q(Be) a) Calculated
MRSDCI
-14.854015 - 1.249639 -0.11541
- 14.937139 - 1.168998 -0.10605
(q, in au) of BeH+ at R=2.4797
(p, in au) and electric field gradients
This work SCF
21 August
LETTERS
1992
au
Ref. [2] HF
- 14.854177 - 1.249255 -0.11551
Ia
SCF
CCSDT-
- 14.853382 - 1.247925 -0.11627
-14.931182 - 1.2151 a) -0.10640
with the CASSCF wavefunction.
Table 2 Variation of q(Be) with the perturbation old for BeH+ at R~2.4797 au. E(SCF)=
energy selection thresh- 14.854015 au
Threshold
No. of CSF
AEcr
q(Be)
1.00x 10-e 0.15x10-6 0.10x 10-e 0.00
5818 9207 9674 68114
-0.082822 -0.083112 -0.083124 -0.083221
-0.10542 -0.10602 -0.10605 -0.10609
the numerical Hartree-Fock values of PyykkS et al. [ 2 1 ] and the calculation of Diercksen and Sadlej [ 21. The data collected in table 1 show clearly that the overall results of this study are superior to those of ref. [ 21; their error in the total energy amounts to 795 uhartree whereas that of the present work is only 162 uhartree; concerning the dipole moment and the electric field gradient at the beryllium nucleus, their errors are 1330 and 760 pau, respectively, whereas Table 3 Electric field gradient
(in au) at the beryllium
and hydrogen
uncertainties about five times smaller, 300 and 100 uau, result in the present work. Concerning now the amount of correlation energy recovered by the final wavefunction, the energy computed at the experimental equilibrium distance in ref. [2], - 14.931182 au, is quite high if compared with our result of - 14.937139 au. The error in the energy, 3.7 kcal/mol relative to our result, is about four times larger than what has been assumed by the theoretical community as a reference value implying chemical accuracy in the calculation. Next was examined the sensitivity of q(Be) to the energy threshold used in the perturbation energy selection of the configuration state functions. The results summarized in table 2 show that q(Be) has practically converged for a threshold in the energy of 1.0x lo-’ hartree. It also shows that the contribution of electron correlation to q(Be) is positive and amounts to 0.00932 au ( z 8Oh).
nuclei in BeH+ as a function
of the internuclear
distance
R
q(Be)
q(H)
R
&Be)
q(H)
1.6 1.7 1.8 2.0 2.1 2.2 2.3 2.4 2.4797 2.5 2.6 2.8 3.0 3.2 3.5
-0.10903 -0.12533 -0.13303 -0.13381 -0.12996 -0.12470 -0.11850 -0.11178 -0.10605 -0.10460 -0.09741 -0.08329 -0.06982 -0.05758 -0.04184
0.98230 0.77982 0.62060 0.39467 0.31493 0.25124 0.2003 1 0.15950 0.13285 0.12672 0.10026 0.06167 0.03640 0.01995 0.00582
3.75 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0
-0.03181 -0.02364 -0.01287 - 0.00708 -0.00388 -0.00213 -0.00058 - 0.00023 - 0.00004 0.00007 0.00000 -0.00001 - 0.00000 - 0.00002
0.00017 -0.00221 -0.00214 -0.00013 0.00107 0.00197 0.00257 0.00220 0.00172 0.00130 0.0004 1 0.00017 0.00009 0.00006
419
Volume 196, number 5
CHEMICAL PHYSICS LETTERS
21 August 1992
BeH+
BeH+
_
0.40
3
O.OO 3 &.J==--
-0.20
1 -0.40
m, 0.00
2.00
4.00
6.00
8.00
R(o.u.) Fig. 1. Electric field gradient at the beryllium and hydrogen nuclei in BeH+ as a function of the internuclear distance.
Fig. 2. Vibrationally averaged electric field gradient in BeH+ as a function of the vibrational quantum number.
The magnitude of q( Be) at the equilibrium internuclear distance computed in this work, -0.10605 au, is a little bit lower than that of ref. [ 2 1,- 0.10640 au, but consistent with their finding that q becomes less negative as the amount of correlation recovered increases. Note also that at the SCF level, the data of ref. [ 21 show that basis set incompleteness in the one-particle space makes q more negative (- 0.11627 au) than the exact numerical Hartree-Fock value of -0.11551 au. Values of the electric field gradient at both the beryllium and hydrogen nuclei as a function of the internuclear distance are collected in table 3 and the
global behavior of these functions is displayed in fig. 1. For hydrogen, q(H) decrease steadily with increasing distance going to zero in the asymptotic limit; a slight minimum is encountered at R z 4.5 au. For beryllium, q(Be) reaches its lowest value at R m 2.0 au and then goes to zero at larger distances. This behavior is very similar to that found by Vojtik et al. [ 131 for the isoelectronic LiH. Vibrationally averaged results of the electric field gradient for selected rotational levels are presented in table 3 together with those computed with the aid of expression (2) of Diercksen and Sadlej [ 21. It is striking to see that the results of ref. [ 21 fail to show
Table 4 Vibrational dependence of the electric field gradient at the Be nucleus computed as a vibrational average and with the aid of expression (2) ofref. [2] V
420
This work
Ref. [2]
N=O
N=l
N=2
N=5
-0.10339 -0.09186 -0.09240 -0.08709 -0.08166 -0.07632 -0.07108 -0.06596 - 0.06093
-0.10336 -0.09782 -0.09237 - 0.08706 -0.08163 -0.07624 -0.07105 -0.06593 - 0.06086
-0.10329 -0.09777 -0.09230 -0.08699 -0.08156 -0.07622 -0.07098 -0.06587 -0.06084
-0.10289 -0.09736 -0.09191 -0.08660 -0.08117 -0.07583 - 0.07060 -0.06549 - 0.06048
-0.10613 -0.10559 -0.10505 -0.10451 -0.10397 -0.10343 -0.10289 -0.10235 -0.10181
Volume 196, number 5
CHEMICAL PHYSICS LETTERS
the strong dependence of q(Be) on the vibrational levels (see also fig. 2 ). In fact, the vibrationally averaged values of q1 and q2 computed in this work already reflect changes of the order of magnitude of the electron correlation corrections. These data clearly show that vibrational corrections introduced via eq. (2) completely spoiled the accuracy implied in the calculation of q. Therefore, the use of that expression and similar ones obtained for other systems do not have the accuracy they imply to have, for the effect of vibrational motion is not properly taken into account. As to the effect of rotation on the value of q, table 4 shows that it is almost negligible.
4. Conclusions This work, by reporting very accurate values of the electric field gradient as a function of the internuclear distance and therefrom computing its vibrational average, shows that Buckingham’s expression is inappropriate for such a correction and that existing calculations on BeH+ and other systems where that formula has been used do not have the accuracy they imply to have.
Acknowledgement The authors are grateful to Professor E.R. Davidson for use of the MELD codes and to Professor W.C. Stwalley for making available a copy of the INTENSITY program. The services of the University of Sgo Paulo Computing Center and of the Data Processing Center of the Instituto de Estudos Avancados are also greatly appreciated. The authors also thank CNPq of Brazil for a research fellowship.
21 August 1992
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