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On the ground-state properties of the germanium dimer
Volume
CHEMSCAL PHYSICS
io7, number I
18 May 1984
LETTERS
ON THE GROUND-STATE PROPERTIES OF THE GE~~~M
DIMER
Gianfranco PACCHIONI Institu t fiir Physikatische Chemie, Freie UniversitoS Berlin. IO00 Berlin 33, Federal Republic of iiennan~ Received
17 February
1984
A theoretical evafuation of the bond length, vibrational frequency and dissociation energy of the Ge2 molecule is reported. The effective core-potential Hartree-Fork cakulations foltoncd by estcnsive Cl give the following v~~Iucs: Re = 4.60 bohr. we_ = 217 cm-l. Dp = 254 eV. Thcsc values arc discussed and compnrcd with those oT previous rheorclical uork and with the znmilable esperimental data.
The ground-state electronic properties of group IVa dimers have been determined expet-imentaiiy with great accuracy with the exception of Ge2, for which only the dissociation energy (De) is known experimentally with certainty [ 1.,221.For this reason, a theoretical prediction of the equilibrium bond length (R,J and the ~bration~ frequency (w,) of the Ge, molecule is of interest. Theoretical investigations indicate 3XS as the molecular ground state of Ge2 [3--S]. This is the experimentaily observed ground state of the isovalent Si2, Snz and Pb, molecules_ However, recent ab initio calculations have shown that, in both Si2 f6] and Ge2 141, the 3X; state is near degenerate with the 3fIu excited state and that a crossing of the two states occurs at the distances 4.24 bohr (Si?) and 4.47 bohr (Ge$, so that for shorter distances 3Jlu is the molecular ground state. The dete~ination of the spectroscopic constants of the Ge2 ground state is still a matter of discussion _ In a recent pseudopotential local-spin-density calculation, Northrup and Cohen [S] studied the electronic structure of the Ge, molecule, obtaining for the 3”; state the following values: R, I= 4.43 bohr, oe = 286 cm- I, De = 4.14 eV. These values differ from our previously reported properties of Ge?. R, = 4.63 bohr, o, = 228 cm- 1 and De = 234 eV 141, computed with a pseudopotential [7] multireference double-excitation configuration-interaction [8] (PP MRD Cf) method. In order to obtain more accurate values for the properties of the Gea ground state. we have perform70
ed extensive PP MRD Cl calculations, adopting a valence CT0 basis set consisting of five primitive s and five primitive p functions contracted to fours and four p and adding two d-polarization functions. This A0 basis set [4s4p2d], of quadruple-zeta quality, has been derived from the basis set II of ref. [3J by adding one diffuse s, one diffuse p and a second d function_ The reference configurations which contribute to the final CI wavefunction by more than 0.3% have been used to generate the Cl space which consists of 17600 configurations. According to the selection procedure of the MRD CT method f8]. only 5000 conjurations have been included in the secular problem (selection threshold T= 5 $artree). The CI energy of the system is then extrapolated to the total CI space (T= 0), while the contribution of higher than single and double excitations to the extrapolated Cl energy has been evaluated according to the Davidson formufa 191. The potentiatenergy curve for the 3Zi ground state of Ge? has been determined in the range 4.44.8 bohr. The values of R,, w, and De computed at SCF and Cl levels are compared in table 1 with the available experimental data. CI gives a bond length, 4.60 bohr, larger than the SCF one (4-51 bohr). Two effects which can in prin-
ciple influence the determination of the equilibrium bond length are not taken into account in our approach: the polarization of the atomic core and relativistic effects. It is well known that the introduction of core polarization in pseudopotential calculations of the equi0 009-2614/84/S 03.00 0 Elsevier Science Publishers (North-Holland Physics publishing Division)
B.V.
Volume
107. number
CHEhllCAL
I
PHYSICS
wtisf3ctov
with Ihc most rcrlenl espsrioi the srabiiiry oi Ge, (D, = 2.65 eV [ I]). Previous esperiments have given even larger values (De = 2.8 eV [Z]). To summarize, our present results give evidence for a Ge-, molecule less strongly bonded than in the calculation of Northrup and Cohen [5]_ The larger equilibrium bond length together with the smaller vibrational frequency and dissociation energy are consistent not only with the previously reported rhearetical investigations [.?.4] but also with the few available esperimental data.
mental RC (bohr)
J) Esrimalcd b)l:rom rrr. librium
bond
full Cl (SW 1~x1). 1151. C) From ref. 111.
bond lengths
influences
of alkali
the results,
distance
which
increases
d) l:rorn
metal
producing
[IO]. However,
number
DC (cv)
WC (WI-’ )
dimers
of the Gel+
of magnitude
smaller
strongly
a shortening
with
ion (0.12
atomic
electric
A3)
dipole
is one order
rhan the corresponding
quantity
for the K+ ion (0.95 A3) [ 1 I J_Therefore. it is rcasonable to expect that the inclusion of core-polarization effects
in the calculation
of the Ge,
bond
length
Bffect
the
rhe 0.1, molecule have given bond shortening between 0.05 and 0.1 bohr [ 13J. which is to be compared with
the 0.064
bohr
relativistic
contraction
of the Cu
However, the relativistic contraction of the Ge 4s orbital is only 0.035 bohr and the corresponding quantity for the 4p orbital is even smaller [ 13,141. For this reason we believe that the shortening of R, caused by relativistic effects will probably not exceed 1% of our computed R,. Finally, it is worth noting that the variational Cl R, of 1.60 bohr practically coincides with the only reported expcrimental determination of the Gel bond length [ 151 (table I). Our computed vibrational frequency, we = 2 17 cm- I, is small not only with respect to the calculation of Northrup and Cohen [ 5 1. but also in comparison with the local-density-functional calculation of Harris and Jones [3] _According to our experience on the isovalent Sn2 and Pb, molecules for which a comparison with experiments is possible. our PP MRD Cl method underestimates the vibrational frequency by about 15% [ 16]_ Therefore. a more realistic prediction is o, = 250 cm- 1. This estimated value compares well with oe = 240 cm- 1 reported by Harris and Jones [3]. Finally, the full Cl estimate of D,, 2.54 eV, is in
4s orbital
[ 13).
The author acknowledges Dr. C.H. Jeung. Dr. J.E. Northrup and Professor M.L. Cohen ior some lnrr‘rcsring discussions and Professor J. Koutecky for a critical reading of rhe nlanussript. The lvork was sup-
ported by DFG-Sonderforschungsbereich turc 2nd Dynamics of Interfaces“.
6 “Slruc-
will
results. More pronounced can be the shortening of R, due to relativistic effects. Recent ab inirio calculations on
not significantly
agreement
measurement
of the
increasing
the theoretical
polarizability
ref. [‘_I.
16 \13)’ 198-l
LETTERS
References J.E. Kingc.tdt Jr.. U.V. Choudary and K;..\. Cingerich. Inor; Chem. I I ( 1979) 3091 121 .A. Krrnr and B.H. Strauss. J. Chcm. Phyh. 45 (1966) 611. R-0. Jones. Phyx Rev. X18 (197s) Zl59. 131 J. Harrisand Mol. Phys. 19 (1983) 777. 141 G. Pacchioni. X1.L. Cohen. Chcm. Phps. Lcr~ers 103 ISI J.E.Korthrupand ( 1953) 110. and R.J. Busnicer. J. Chum. 161 P. Bruna. S.D. Pcyrrimhuii Pliys. 72 t 1980) 5l37. Thlrore1 Chim Act3 35 171 Ph. Durand and J.C. BJrrhelJr. (1975) 283. Chum. IS1 R.J. Bucnker and S.D. Pcycrinlhor’i. Throw. Acra 35 (1971) 33. in: The \vorU ot’qux~rum chcmirrr!. 191 E.R. Daidson, cds. R. Daudsl znd B. Pullmann (Rad?l. Derdrrchr. 197:). G.H. Jan:. J.P. \l;ilriw snd J.P. DAudcy. J. Chcm. 1101 Phys. 77 (I 982) 337 1. and K_\l.S. SA..znz. HAndbook II 11 S. Fmge. J. liiclrwo~vky OI atomic d3t.t (Elwvisr. AmsrcrdJm. 1976). R.L. Marin. J.Chrm. Phys. 75 11963) 5Sll; 1111 Xl. P&.sicr. J. Chum. Phys. 79 (19S3) 2099; H. Srull. P. I’ucntealb3. Xl. Do!:. J. Fl>d. L. van Srcn~p_ily and H. Prcuss. J. Chrrn. Phys. 79 (1983) 5532. J .P. DrscLtu\. XI. Dam Xucl. Dar3 TJblcs I2 (I 973) 1131 311. Il4i 1V.H.E. Sclr~\urr, S.Y. Chu 2nd F. Clark. .51al. Phys. 50
III
(1983) 603. 1 IS] G.V. Cadigzk. Y.N. Morokov. X.C. \lukhx.h~v and S.V. Chernov. 211. Srrukl. Khini. “(5) (1981) 36. and references thcrcin. [ 161 C. Pxchioni, unpublished resulr.
71
CHEMSCAL PHYSICS
io7, number I
18 May 1984
LETTERS
ON THE GROUND-STATE PROPERTIES OF THE GE~~~M
DIMER
Gianfranco PACCHIONI Institu t fiir Physikatische Chemie, Freie UniversitoS Berlin. IO00 Berlin 33, Federal Republic of iiennan~ Received
17 February
1984
A theoretical evafuation of the bond length, vibrational frequency and dissociation energy of the Ge2 molecule is reported. The effective core-potential Hartree-Fork cakulations foltoncd by estcnsive Cl give the following v~~Iucs: Re = 4.60 bohr. we_ = 217 cm-l. Dp = 254 eV. Thcsc values arc discussed and compnrcd with those oT previous rheorclical uork and with the znmilable esperimental data.
The ground-state electronic properties of group IVa dimers have been determined expet-imentaiiy with great accuracy with the exception of Ge2, for which only the dissociation energy (De) is known experimentally with certainty [ 1.,221.For this reason, a theoretical prediction of the equilibrium bond length (R,J and the ~bration~ frequency (w,) of the Ge, molecule is of interest. Theoretical investigations indicate 3XS as the molecular ground state of Ge2 [3--S]. This is the experimentaily observed ground state of the isovalent Si2, Snz and Pb, molecules_ However, recent ab initio calculations have shown that, in both Si2 f6] and Ge2 141, the 3X; state is near degenerate with the 3fIu excited state and that a crossing of the two states occurs at the distances 4.24 bohr (Si?) and 4.47 bohr (Ge$, so that for shorter distances 3Jlu is the molecular ground state. The dete~ination of the spectroscopic constants of the Ge2 ground state is still a matter of discussion _ In a recent pseudopotential local-spin-density calculation, Northrup and Cohen [S] studied the electronic structure of the Ge, molecule, obtaining for the 3”; state the following values: R, I= 4.43 bohr, oe = 286 cm- I, De = 4.14 eV. These values differ from our previously reported properties of Ge?. R, = 4.63 bohr, o, = 228 cm- 1 and De = 234 eV 141, computed with a pseudopotential [7] multireference double-excitation configuration-interaction [8] (PP MRD Cf) method. In order to obtain more accurate values for the properties of the Gea ground state. we have perform70
ed extensive PP MRD Cl calculations, adopting a valence CT0 basis set consisting of five primitive s and five primitive p functions contracted to fours and four p and adding two d-polarization functions. This A0 basis set [4s4p2d], of quadruple-zeta quality, has been derived from the basis set II of ref. [3J by adding one diffuse s, one diffuse p and a second d function_ The reference configurations which contribute to the final CI wavefunction by more than 0.3% have been used to generate the Cl space which consists of 17600 configurations. According to the selection procedure of the MRD CT method f8]. only 5000 conjurations have been included in the secular problem (selection threshold T= 5 $artree). The CI energy of the system is then extrapolated to the total CI space (T= 0), while the contribution of higher than single and double excitations to the extrapolated Cl energy has been evaluated according to the Davidson formufa 191. The potentiatenergy curve for the 3Zi ground state of Ge? has been determined in the range 4.44.8 bohr. The values of R,, w, and De computed at SCF and Cl levels are compared in table 1 with the available experimental data. CI gives a bond length, 4.60 bohr, larger than the SCF one (4-51 bohr). Two effects which can in prin-
ciple influence the determination of the equilibrium bond length are not taken into account in our approach: the polarization of the atomic core and relativistic effects. It is well known that the introduction of core polarization in pseudopotential calculations of the equi0 009-2614/84/S 03.00 0 Elsevier Science Publishers (North-Holland Physics publishing Division)
B.V.
Volume
107. number
CHEhllCAL
I
PHYSICS
wtisf3ctov
with Ihc most rcrlenl espsrioi the srabiiiry oi Ge, (D, = 2.65 eV [ I]). Previous esperiments have given even larger values (De = 2.8 eV [Z]). To summarize, our present results give evidence for a Ge-, molecule less strongly bonded than in the calculation of Northrup and Cohen [5]_ The larger equilibrium bond length together with the smaller vibrational frequency and dissociation energy are consistent not only with the previously reported rhearetical investigations [.?.4] but also with the few available esperimental data.
mental RC (bohr)
J) Esrimalcd b)l:rom rrr. librium
bond
full Cl (SW 1~x1). 1151. C) From ref. 111.
bond lengths
influences
of alkali
the results,
distance
which
increases
d) l:rorn
metal
producing
[IO]. However,
number
DC (cv)
WC (WI-’ )
dimers
of the Gel+
of magnitude
smaller
strongly
a shortening
with
ion (0.12
atomic
electric
A3)
dipole
is one order
rhan the corresponding
quantity
for the K+ ion (0.95 A3) [ 1 I J_Therefore. it is rcasonable to expect that the inclusion of core-polarization effects
in the calculation
of the Ge,
bond
length
Bffect
the
rhe 0.1, molecule have given bond shortening between 0.05 and 0.1 bohr [ 13J. which is to be compared with
the 0.064
bohr
relativistic
contraction
of the Cu
However, the relativistic contraction of the Ge 4s orbital is only 0.035 bohr and the corresponding quantity for the 4p orbital is even smaller [ 13,141. For this reason we believe that the shortening of R, caused by relativistic effects will probably not exceed 1% of our computed R,. Finally, it is worth noting that the variational Cl R, of 1.60 bohr practically coincides with the only reported expcrimental determination of the Gel bond length [ 151 (table I). Our computed vibrational frequency, we = 2 17 cm- I, is small not only with respect to the calculation of Northrup and Cohen [ 5 1. but also in comparison with the local-density-functional calculation of Harris and Jones [3] _According to our experience on the isovalent Sn2 and Pb, molecules for which a comparison with experiments is possible. our PP MRD Cl method underestimates the vibrational frequency by about 15% [ 16]_ Therefore. a more realistic prediction is o, = 250 cm- 1. This estimated value compares well with oe = 240 cm- 1 reported by Harris and Jones [3]. Finally, the full Cl estimate of D,, 2.54 eV, is in
4s orbital
[ 13).
The author acknowledges Dr. C.H. Jeung. Dr. J.E. Northrup and Professor M.L. Cohen ior some lnrr‘rcsring discussions and Professor J. Koutecky for a critical reading of rhe nlanussript. The lvork was sup-
ported by DFG-Sonderforschungsbereich turc 2nd Dynamics of Interfaces“.
6 “Slruc-
will
results. More pronounced can be the shortening of R, due to relativistic effects. Recent ab inirio calculations on
not significantly
agreement
measurement
of the
increasing
the theoretical
polarizability
ref. [‘_I.
16 \13)’ 198-l
LETTERS
References J.E. Kingc.tdt Jr.. U.V. Choudary and K;..\. Cingerich. Inor; Chem. I I ( 1979) 3091 121 .A. Krrnr and B.H. Strauss. J. Chcm. Phyh. 45 (1966) 611. R-0. Jones. Phyx Rev. X18 (197s) Zl59. 131 J. Harrisand Mol. Phys. 19 (1983) 777. 141 G. Pacchioni. X1.L. Cohen. Chcm. Phps. Lcr~ers 103 ISI J.E.Korthrupand ( 1953) 110. and R.J. Busnicer. J. Chum. 161 P. Bruna. S.D. Pcyrrimhuii Pliys. 72 t 1980) 5l37. Thlrore1 Chim Act3 35 171 Ph. Durand and J.C. BJrrhelJr. (1975) 283. Chum. IS1 R.J. Bucnker and S.D. Pcycrinlhor’i. Throw. Acra 35 (1971) 33. in: The \vorU ot’qux~rum chcmirrr!. 191 E.R. Daidson, cds. R. Daudsl znd B. Pullmann (Rad?l. Derdrrchr. 197:). G.H. Jan:. J.P. \l;ilriw snd J.P. DAudcy. J. Chcm. 1101 Phys. 77 (I 982) 337 1. and K_\l.S. SA..znz. HAndbook II 11 S. Fmge. J. liiclrwo~vky OI atomic d3t.t (Elwvisr. AmsrcrdJm. 1976). R.L. Marin. J.Chrm. Phys. 75 11963) 5Sll; 1111 Xl. P&.sicr. J. Chum. Phys. 79 (19S3) 2099; H. Srull. P. I’ucntealb3. Xl. Do!:. J. Fl>d. L. van Srcn~p_ily and H. Prcuss. J. Chrrn. Phys. 79 (1983) 5532. J .P. DrscLtu\. XI. Dam Xucl. Dar3 TJblcs I2 (I 973) 1131 311. Il4i 1V.H.E. Sclr~\urr, S.Y. Chu 2nd F. Clark. .51al. Phys. 50
III
(1983) 603. 1 IS] G.V. Cadigzk. Y.N. Morokov. X.C. \lukhx.h~v and S.V. Chernov. 211. Srrukl. Khini. “(5) (1981) 36. and references thcrcin. [ 161 C. Pxchioni, unpublished resulr.
71