- Email: [email protected]
The atomization energy of Mg4
5 February 1999
Chemical Physics Letters 300 Ž1999. 364–368
The atomization energy of Mg 4 Charles W. Bauschlicher Jr. ) , Harry Partridge Mail Stop-230-3, NASA Ames Research Center, Moffett Field, CA 94035, USA Received 10 November 1998
Abstract The atomization energy of Mg 4 is determined using the MP2 and CCSDŽT. levels of theory. Basis set incompleteness, basis set extrapolation, and core-valence effects are discussed. Our best atomization energy, including the zero-point energy and scalar relativistic effects, is 24.6 " 1.6 kcalrmol. Our computed and extrapolated values are compared with previous results, where it is observed that our extrapolated MP2 value is in good agreement with the MP2-R12 value. The CCSDŽT. and MP2 core effects are found to have the opposite signs. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction It has been suggested that properly positioned Mg atoms on a hydrogen passivated SiŽ111. surface could function as an electronic device w1x. To help determine the feasibility of this idea, we have undertaken studies w2x of Mg mobility on the surface, as well as, the study of Mg–Mg interactions. As part of this study, we reinvestigated the binding energy of Mg 4 , which is reported in this manuscript. Although the atomization energy of Mg 4 has been the subject of many studies, three of these are particularly noteworthy for determining an accurate atomization energy. The first two are by Lee, Rendell, and Taylor w3x ŽLRT., where the multireference configuration interaction ŽMRCI. and coupled cluster singles and doubles w4x, with a perturbational estimate of the triples w5x ŽCCSDŽT.., approaches are used. The third significant contribution is that of Klopper and Almlof ¨ w6x, where second-order Møller–Plesset perturbation theory ŽMP2. is used to )
Corresponding author. E-mail: [email protected]
explore the question of basis set completeness. They used both the conventional MP2 approach, as well as, one including linear terms in the interelectronic distance, MP2-R12. In addition to exploring the basis set completeness, they also used the MP2-R12 approach to estimate the contribution made by the Mg 2s and 2p core electrons to the atomization energy; the calculation of core and core-valence ŽCV. correlation effects is known to be very difficult. Using their MP2 results along with the results obtained by LRT, Klopper and Almlof ¨ made a revised estimate for the atomization energy. We note that using MP2 results to estimate the basis set incompleteness is very common w7,8x. One significant finding of Klopper and Almlof ¨ was how far even large basis set MP2 results were from the basis set limit. Since the publication of these three studies, there has been extensive work on the development w9–12x of systematic basis sets and on extrapolating the results to the complete basis set limit w13,14x. We therefore investigate how well the conventional MP2 approach with and without extrapolation reproduces
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 3 8 1 - 5
C.W. Bauschlicher Jr., H. Partridger Chemical Physics Letters 300 (1999) 364–368
365
the MP2-R12 result. It is also possible to extrapolate the CCSDŽT. results and to compute the CV effect at the CCSDŽT. level. Thus we are able to address how additive are the MP2 corrections for this system.
tions are performed using Molpro97 w20x. The DK calculations are performed using MOLECULESWEDEN w21x, TITAN w22x, and the DK integral program of Hess w18x.
2. Methods
3. Results and discussion
The hybrid-B3LYP approach w15,16x is used in conjunction with the 6-311G ) basis set w17x. The MP2 and CCSDŽT. approaches w4,5x are used in conjunction with the correlation consistent polarized valence Žcc-pV. sets of Woon and Dunning w12x; the triple zeta ŽTZ., quadruple zeta ŽQZ., and quintuple zeta Ž5Z. sets are used. In a second series of valence calculations, extra flexibility is introduced into the cc-pV basis sets by adding even-tempered diffuse functions; an spd set is added to the TZ basis, an spdf set to the QZ basis, and an spdfg set to the 5Z basis. In addition, two even-tempered Žwith a b of 3. tight d functions are added to each set. Finally the sp contraction is changed; for the TZ, QZ, and 5Z sets, the inner 10s4p, 11s5p, and 15s6p functions, respectively, are contracted to two s and one p function, with all of the remaining functions uncontracted. We denote these enhanced basis sets as TZX , QZX , and 5ZX , or collectively as cc-pVX . A third basis set ŽCV basis. is developed to study the effect of Mg 2s2p correlation. Starting from the cc-pVQZ set, the sp contraction is changed; the inner 11 s functions are contracted to three functions and the inner four p functions to one function, with the outer functions uncontracted. Two tight d and two tight f functions are added with exponents of 2.23 and 0.893. To estimate the complete basis set limit, several extrapolation techniques are used. We use the twopoint ny3 scheme described by Helgaker et al. w13x. We also use the two-point ny4 , three-point Ž ny4 q ny6 ., and variable a Ž nya . schemes described by Martin w14x. The scalar relativistic effect is computed using the one-electron Douglas Kroll ŽDK. approach w18x. In the DK calculations, the cc-pVTZX basis set is contracted to the same size as in the nonrelativistic calculations, but the contraction coefficients are taken from a DK Mg atomic calculation. The B3LYP calculations are performed using Gaussian94 w19x. The MP2 and CCSDŽT. calcula-
We optimize the geometry at the B3LYPr6311G ) , CCSDŽT.rcc-pVTZ, and CCSDŽT.rccpVQZ levels of theory; the Mg–Mg distances are ˚ respectively. The geome3.172, 3.103, and 3.095 A, try is not optimized using the cc-pV5Z basis set because of computational expense, but from the small difference between the cc-pVTZ and cc-pVQZ sets, we suspect that the cc-pVQZ result is nearly converged Žat least for a valence electron treatment.. The B3LYP Mg–Mg distance is longer than the CCSDŽT. values. Expanding the basis set to 6-311 q ˚ thus GŽ2df. reduces the B3LYP value to 3.162 A, unlike many other systems, improving the basis set does not significantly improve the B3LYP bond length. Our CCSDŽT. Mg–Mg distances are smaller than that found by LRT at the CCSDŽT. level Ž3.110 ˚ ., because we use superior basis sets. A The B3LYPr6-311G ) harmonic frequencies of 129Ž e ., 146Ž te ., and 157Ž a1 . cmy1 are in reasonable agreement with the CCSDŽT. values of LRT, which are 147, 171, and 192 cmy1 . These harmonic frequencies lead to a zero-point energy of 1.22 and 1.43 kcalrmol for the B3LYP and CCSDŽT. levels, respectively. Expanding the basis set does not significantly improve the B3LYP harmonic frequencies. The B3LYPr6-311G ) atomization energy of 12.2 kcalrmol is much smaller than that obtained using the other methods and is not discussed further. The MP2 and CCSDŽT. atomization energies computed using the DFT geometry are reported in Table 1 and the extrapolated values in Table 2. The atomization energies using the CCSDŽT.rcc-pVTZ geometry are also reported. Using the CCSDŽT.rcc-pVTZ geometry increases the extrapolated MP2 values by about 1 kcalrmol, while the CCSDŽT. values increase by only about 0.4 kcalrmol. Correcting for BSSE does not significantly affect the CCSDŽT. extrapolated values, while the MP2 values are reduced by about 0.2 kcalrmol. Thus the MP2 and CCSDŽT. results show slightly different effects of the geometry change and of the BSSE correction.
C.W. Bauschlicher Jr., H. Partridger Chemical Physics Letters 300 (1999) 364–368
366
Table 1 Summary of the computed Mg 4 atomization energies, in kcalrmol Basis
SCF
MP2
CCSDŽT.
˚ cc-pV basis at B3LYPr6-311G ) re s 3.1719 A TZ y7.580 26.897 23.240 QZ y7.474 28.760 25.334 5Z y7.340 29.494 25.985 cc-pV basis at CCSDŽT.rcc-pVTZ re s 3.1025 a TZ y9.108 27.620 23.492 TZ-BSSE y9.135 26.955 23.286 QZ y8.983 29.571 25.645 QZ-BSSE y8.993 29.154 25.565 5Z y8.826 30.365 26.339 CV basis at re s 3.1025 CVŽval. y8.787 29.988 26.045 CVŽcv. 30.417 25.107 Db 0.429 y0.938 X cc-pV basis sets at re s 3.1025 TZ y8.787 28.599 24.525 TZŽDK. y9.092 28.111 24.131 QZ y8.776 30.173 26.232 5Z y8.756 30.690 26.674 X cc-pV basis sets at re s 3.110 TZ y8.604 28.535 24.514 ˚ previous results c at re s 3.110 A w7631x d y9.04 27.0 big y8.60 28.1 R12 31.4 R12Žcv. 31.6 D 0.2 a b c d e
23.9 e
˚ The CCSDŽT.rcc-pVQZ re value is 3.0949A. The core-valence contribution to the atomization energy. Taken from Klopper and Almlof ¨ w6x unless otherwise noted. See Klopper and Almlof ¨ for the definition of their basis sets. Taken from Lee, Rendell, and Taylor w3x.
Using the CV basis set, calculations are performed that correlate the Mg 2s and 2p electrons as well as the 3s electrons. At the MP2 level, correlating the inner-shell electrons increases the atomization energy by 0.43 kcalrmol, but at the CCSDŽT. level, the atomization energy is decreased by 0.94 kcalrmol. This difference demonstrates that the MP2 level will not always yield a reliable inner-shell effect. Using the CV basis set increases the valence CCSDŽT. atomization energy by 0.4 kcalrmol compared with the cc-pVQZ basis set from which it is derived, see Table 1. Half of this differences appears at the SCF level. We therefore repeat the MP2 and
CCSDŽT. calculations using our enhanced cc-pVX basis set. The increase in atomization between the cc-pV and cc-pVX basis sets is the largest for the TZ basis set, and decreases in magnitude as the basis set increases in size. The effect of enhancing the basis set is less than 0.1 kcalrmol on the 3-point extrapolated CCSDŽT. values. At the MP2 level, the ny4 q ny6 extrapolated value does not change when the basis set is improved from cc-pV to cc-pVX , while the MP2 value obtained using the variable a method actually decreases when the basis set is improved. Our experience w23x has been that the most reliable atomization energies are obtained for the basis set series with the most consistent results for the different extrapolation approaches. Thus we take our extrapolated value of 31.1 kcalrmol as our best estimate of the complete basis set MP2 result, which is similar to the 31.4 kcalrmol MP2-R12 value of Klopper and Almlof. ¨ Our best Žcc-pV5ZX . computed value of 30.7 kcalrmol is much better than the MP2 value Klopper and Almlof ¨ obtained with their big basis set. This again confirms the accuracy of the modern cc-pV basis sets. The relativistic effect on the atomization energy is y0.39 kcalrmol, see Table 1. This is small, but not insignificant. It is difficult to assign an error to this quantity, so we take a conservative approach and assume that it could be in error by "0.20 kcalrmol. We estimate our best atomization energy as follows. We take the ny4 q ny6 ŽTQ5. CCSDŽT. value from the cc-pVX enhanced basis set as our best value. We assume this extrapolated value is within about 0.5 kcalrmol of the CBS value on the basis of the difference between our extrapolated MP2 value using the enhanced basis set Ž31.1 kcalrmol. and the MP2-R12 value of Klopper and Almlof ¨ Ž31.4 kcalrmol.. Also the difference between our cc-pV5ZX value Ž26.67 kcalrmol. and the extrapolated value Ž26.97 kcalrmol. is only 0.3 kcalrmol, and the extrapolation should be accurate to within a factor of two. LRT found that the CCSDŽT. value was 0.2 kcalrmol smaller than the MRCI value in a small basis set that only recovered about half the binding energy. We therefore assume that the effect of higher levels of correlation could be twice that found by LRT, and thus correct our atomization energy by 0.4 " 0.4 kcalrmol for higher level correlation effects. We correct the atomization energy using the
C.W. Bauschlicher Jr., H. Partridger Chemical Physics Letters 300 (1999) 364–368
367
Table 2 Summary of extrapolated Mg 4 atomization energies, in kcalrmol Method
a
ny3 TQ
ny3 Q5
˚ cc-pV basis at B3LYPr6-311G ) re s 3.1719 A MP2 30.12 30.26 CCSDŽT. 26.86 26.67 cc-pV basis at CCSDŽT.rcc-pVTZ re s 3.1025 MP2 30.99 31.20 MP2–BSSE 30.76 CCSDŽT. 27.22 27.07 CCSDŽT. –BSSE 27.23 X cc-pV basis sets at re s 3.1025 MP2 31.32 31.23 CCSDŽT. 27.48 27.14 a b
ny4 TQ
ny4 Q5
ny4 q ny6 TQ5
a b TQ5
29.84 26.54
30.09 26.51
30.18 26.50
30.35Ž3.086. 26.49Ž4.104.
30.70 30.42 26.89 26.88
31.01
31.12
31.35Ž2.946.
26.90
26.91
26.91Ž3.949.
31.08 27.22
31.11 27.03
31.12 26.97
31.13Ž3.868. 26.94Ž4.884.
The extrapolation method is given in the first line and the basis sets used in the second. The alpha is given in parentheses.
CCSDŽT. core-valence effect of y0.94 kcalrmol. It is difficult to estimate the error in the core-valence contribution, but we assume that it can be as large as 50% of the computed effect. Thus our final De value is 26.97 " 0.5 y 0.94 " 0.47 q 0.4 " 0.4 s 26.4 " 1.4 kcalrmol, assuming that the error estimates are additive. This value is similar to that of Klopper and Almlof ¨ Ž27.8 kcalrmol., but smaller than the less rigorous estimate of LRT Ž31 kcalrmol.. We note that the Klopper and Almlof ¨ valence value is 27.6 kcalrmol, which is very similar to our valence estimate of 27.4 " 0.9 kcalrmol; the main difference between our best estimate and that of Klopper and Almlof ¨ is a result of the difference in their MP2 Ž0.2 kcalrmol. and our CCSDŽT. Žy0.94 kcalrmol. core effect. To obtain our final value, we include the zeropoint energy and scalar relativistic effects, which were not considered by Klopper and Almlof. ¨ Using our computed DK scalar relativistic effect and the CCSDŽT. zero-point energy w3x, yields our best estimate of the Mg 4 atomization energy of 24.6 " 1.6 kcalrmol.
4. Conclusions Our best MP2 value is in good agreement with the MP2-R12 value of Klopper and Almlof. ¨ Our extrapolated MP2 value is only 0.3 kcalrmol smaller than
the MP2-R12 value. The core-valence effect at the CCSDŽT. level has the opposite sign as determined at the MP2 level. Our best atomization energy, including zero-point energy and scalar relativistic effects, is 24.6 " 1.6 kcalrmol. References w1x T. Yamada, J. Vac. Sci. Technol. A. 16 Ž1998. 1403. w2x T. Yamada, C.W. Bauschlicher, H. Partridge, Phys. Rev. B., submitted. w3x T.J. Lee, A.P. Rendell, P.R. Taylor, J. Chem. Phys. 92 Ž1990. 489; 93 Ž1990. 6636. w4x R.J. Bartlett, Annu. Rev. Phys. Chem. 32 Ž1981. 359. w5x K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157 Ž1989. 479. w6x W. Klopper, J. Almlof, ¨ J. Chem. Phys. 99 Ž1993. 5167. w7x L.A. Curtiss, K. Raghavachari, G.W. Trucks, J.A. Pople, J. Chem. Phys. 94 Ž1991. 7221. w8x A. Ricca, C.W. Bauschlicher, J. Phys. Chem. 102 Ž1998. 876. w9x T.H. Dunning, J. Chem. Phys. 90 Ž1989. 1007. w10x R.A. Kendall, T.H. Dunning, R.J. Harrison, J. Chem. Phys. 96 Ž1992. 6796. w11x D.E. Woon, T.H. Dunning, J. Chem. Phys. 98 Ž1993. 1358. w12x D.E. Woon, T.H. Dunning Jr., to be published. w13x T. Helgaker, W. Klopper, H. Koch, J. Noga, J. Chem. Phys. 106 Ž1997. 9639. w14x J.M.L. Martin, Chem. Phys. Lett. 259 Ž1996. 669. w15x A.D. Becke, J. Chem. Phys. 98 Ž1993. 5648. w16x P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chem. 98 Ž1994. 11623. w17x M.J. Frisch, J.A. Pople, J.S. Binkley, J. Chem. Phys. 80 Ž1984. 3265, and references therein.
368
C.W. Bauschlicher Jr., H. Partridger Chemical Physics Letters 300 (1999) 364–368
w18x B.A. Hess, Phys. Rev. A 32 Ž1985. 756. w19x Gaussian 94, Revision D.1, M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, Gaussian, Inc., Pittsburgh PA, 1995. w20x MOLPRO is a package of ab initio programs written by H.-J. Werner, P.J. Knowles, with contributions from J. Almlof, ¨ R.D. Amos, A. Berning, D.L. Cooper, M.J.O. Deegan, A.J.
w21x
w22x w23x
Dobbyn, F. Eckert, S.T. Elbert, C. Hampel, R. Lindh, A.W. Llyod, W. Meyer, A. Nicklass, K. Peterson, R. Pitzer, A.J. Stone, P.R. Taylor, M.E. Mura, P. Pulay, M. Schutz, ¨ H. Stoll, T. Thorseinsson. The closed shell CCSD program is described in C. Hampel, K. Peterson, H.-J. Werner, Chem. Phys. Lett. 190 Ž1992. 1. MOLECULE-SWEDEN is an electronic structure program written by Almlof ¨ J, Bauschlicher CW, Blomberg MRA, ˚ RenChong DP, Heiberg A, Langhoff SR, Malmqvist P-A, dell AP, Roos BO, Siegbahn PEM, Taylor PR. TITAN is a set of electronic structure programs written by T.J. Lee, A.P. Rendell, J.E. Rice. C.W. Bauschlicher, A. Ricca, J. Phys. Chem. 102 Ž1998. 8044.
Chemical Physics Letters 300 Ž1999. 364–368
The atomization energy of Mg 4 Charles W. Bauschlicher Jr. ) , Harry Partridge Mail Stop-230-3, NASA Ames Research Center, Moffett Field, CA 94035, USA Received 10 November 1998
Abstract The atomization energy of Mg 4 is determined using the MP2 and CCSDŽT. levels of theory. Basis set incompleteness, basis set extrapolation, and core-valence effects are discussed. Our best atomization energy, including the zero-point energy and scalar relativistic effects, is 24.6 " 1.6 kcalrmol. Our computed and extrapolated values are compared with previous results, where it is observed that our extrapolated MP2 value is in good agreement with the MP2-R12 value. The CCSDŽT. and MP2 core effects are found to have the opposite signs. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction It has been suggested that properly positioned Mg atoms on a hydrogen passivated SiŽ111. surface could function as an electronic device w1x. To help determine the feasibility of this idea, we have undertaken studies w2x of Mg mobility on the surface, as well as, the study of Mg–Mg interactions. As part of this study, we reinvestigated the binding energy of Mg 4 , which is reported in this manuscript. Although the atomization energy of Mg 4 has been the subject of many studies, three of these are particularly noteworthy for determining an accurate atomization energy. The first two are by Lee, Rendell, and Taylor w3x ŽLRT., where the multireference configuration interaction ŽMRCI. and coupled cluster singles and doubles w4x, with a perturbational estimate of the triples w5x ŽCCSDŽT.., approaches are used. The third significant contribution is that of Klopper and Almlof ¨ w6x, where second-order Møller–Plesset perturbation theory ŽMP2. is used to )
Corresponding author. E-mail: [email protected]
explore the question of basis set completeness. They used both the conventional MP2 approach, as well as, one including linear terms in the interelectronic distance, MP2-R12. In addition to exploring the basis set completeness, they also used the MP2-R12 approach to estimate the contribution made by the Mg 2s and 2p core electrons to the atomization energy; the calculation of core and core-valence ŽCV. correlation effects is known to be very difficult. Using their MP2 results along with the results obtained by LRT, Klopper and Almlof ¨ made a revised estimate for the atomization energy. We note that using MP2 results to estimate the basis set incompleteness is very common w7,8x. One significant finding of Klopper and Almlof ¨ was how far even large basis set MP2 results were from the basis set limit. Since the publication of these three studies, there has been extensive work on the development w9–12x of systematic basis sets and on extrapolating the results to the complete basis set limit w13,14x. We therefore investigate how well the conventional MP2 approach with and without extrapolation reproduces
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 3 8 1 - 5
C.W. Bauschlicher Jr., H. Partridger Chemical Physics Letters 300 (1999) 364–368
365
the MP2-R12 result. It is also possible to extrapolate the CCSDŽT. results and to compute the CV effect at the CCSDŽT. level. Thus we are able to address how additive are the MP2 corrections for this system.
tions are performed using Molpro97 w20x. The DK calculations are performed using MOLECULESWEDEN w21x, TITAN w22x, and the DK integral program of Hess w18x.
2. Methods
3. Results and discussion
The hybrid-B3LYP approach w15,16x is used in conjunction with the 6-311G ) basis set w17x. The MP2 and CCSDŽT. approaches w4,5x are used in conjunction with the correlation consistent polarized valence Žcc-pV. sets of Woon and Dunning w12x; the triple zeta ŽTZ., quadruple zeta ŽQZ., and quintuple zeta Ž5Z. sets are used. In a second series of valence calculations, extra flexibility is introduced into the cc-pV basis sets by adding even-tempered diffuse functions; an spd set is added to the TZ basis, an spdf set to the QZ basis, and an spdfg set to the 5Z basis. In addition, two even-tempered Žwith a b of 3. tight d functions are added to each set. Finally the sp contraction is changed; for the TZ, QZ, and 5Z sets, the inner 10s4p, 11s5p, and 15s6p functions, respectively, are contracted to two s and one p function, with all of the remaining functions uncontracted. We denote these enhanced basis sets as TZX , QZX , and 5ZX , or collectively as cc-pVX . A third basis set ŽCV basis. is developed to study the effect of Mg 2s2p correlation. Starting from the cc-pVQZ set, the sp contraction is changed; the inner 11 s functions are contracted to three functions and the inner four p functions to one function, with the outer functions uncontracted. Two tight d and two tight f functions are added with exponents of 2.23 and 0.893. To estimate the complete basis set limit, several extrapolation techniques are used. We use the twopoint ny3 scheme described by Helgaker et al. w13x. We also use the two-point ny4 , three-point Ž ny4 q ny6 ., and variable a Ž nya . schemes described by Martin w14x. The scalar relativistic effect is computed using the one-electron Douglas Kroll ŽDK. approach w18x. In the DK calculations, the cc-pVTZX basis set is contracted to the same size as in the nonrelativistic calculations, but the contraction coefficients are taken from a DK Mg atomic calculation. The B3LYP calculations are performed using Gaussian94 w19x. The MP2 and CCSDŽT. calcula-
We optimize the geometry at the B3LYPr6311G ) , CCSDŽT.rcc-pVTZ, and CCSDŽT.rccpVQZ levels of theory; the Mg–Mg distances are ˚ respectively. The geome3.172, 3.103, and 3.095 A, try is not optimized using the cc-pV5Z basis set because of computational expense, but from the small difference between the cc-pVTZ and cc-pVQZ sets, we suspect that the cc-pVQZ result is nearly converged Žat least for a valence electron treatment.. The B3LYP Mg–Mg distance is longer than the CCSDŽT. values. Expanding the basis set to 6-311 q ˚ thus GŽ2df. reduces the B3LYP value to 3.162 A, unlike many other systems, improving the basis set does not significantly improve the B3LYP bond length. Our CCSDŽT. Mg–Mg distances are smaller than that found by LRT at the CCSDŽT. level Ž3.110 ˚ ., because we use superior basis sets. A The B3LYPr6-311G ) harmonic frequencies of 129Ž e ., 146Ž te ., and 157Ž a1 . cmy1 are in reasonable agreement with the CCSDŽT. values of LRT, which are 147, 171, and 192 cmy1 . These harmonic frequencies lead to a zero-point energy of 1.22 and 1.43 kcalrmol for the B3LYP and CCSDŽT. levels, respectively. Expanding the basis set does not significantly improve the B3LYP harmonic frequencies. The B3LYPr6-311G ) atomization energy of 12.2 kcalrmol is much smaller than that obtained using the other methods and is not discussed further. The MP2 and CCSDŽT. atomization energies computed using the DFT geometry are reported in Table 1 and the extrapolated values in Table 2. The atomization energies using the CCSDŽT.rcc-pVTZ geometry are also reported. Using the CCSDŽT.rcc-pVTZ geometry increases the extrapolated MP2 values by about 1 kcalrmol, while the CCSDŽT. values increase by only about 0.4 kcalrmol. Correcting for BSSE does not significantly affect the CCSDŽT. extrapolated values, while the MP2 values are reduced by about 0.2 kcalrmol. Thus the MP2 and CCSDŽT. results show slightly different effects of the geometry change and of the BSSE correction.
C.W. Bauschlicher Jr., H. Partridger Chemical Physics Letters 300 (1999) 364–368
366
Table 1 Summary of the computed Mg 4 atomization energies, in kcalrmol Basis
SCF
MP2
CCSDŽT.
˚ cc-pV basis at B3LYPr6-311G ) re s 3.1719 A TZ y7.580 26.897 23.240 QZ y7.474 28.760 25.334 5Z y7.340 29.494 25.985 cc-pV basis at CCSDŽT.rcc-pVTZ re s 3.1025 a TZ y9.108 27.620 23.492 TZ-BSSE y9.135 26.955 23.286 QZ y8.983 29.571 25.645 QZ-BSSE y8.993 29.154 25.565 5Z y8.826 30.365 26.339 CV basis at re s 3.1025 CVŽval. y8.787 29.988 26.045 CVŽcv. 30.417 25.107 Db 0.429 y0.938 X cc-pV basis sets at re s 3.1025 TZ y8.787 28.599 24.525 TZŽDK. y9.092 28.111 24.131 QZ y8.776 30.173 26.232 5Z y8.756 30.690 26.674 X cc-pV basis sets at re s 3.110 TZ y8.604 28.535 24.514 ˚ previous results c at re s 3.110 A w7631x d y9.04 27.0 big y8.60 28.1 R12 31.4 R12Žcv. 31.6 D 0.2 a b c d e
23.9 e
˚ The CCSDŽT.rcc-pVQZ re value is 3.0949A. The core-valence contribution to the atomization energy. Taken from Klopper and Almlof ¨ w6x unless otherwise noted. See Klopper and Almlof ¨ for the definition of their basis sets. Taken from Lee, Rendell, and Taylor w3x.
Using the CV basis set, calculations are performed that correlate the Mg 2s and 2p electrons as well as the 3s electrons. At the MP2 level, correlating the inner-shell electrons increases the atomization energy by 0.43 kcalrmol, but at the CCSDŽT. level, the atomization energy is decreased by 0.94 kcalrmol. This difference demonstrates that the MP2 level will not always yield a reliable inner-shell effect. Using the CV basis set increases the valence CCSDŽT. atomization energy by 0.4 kcalrmol compared with the cc-pVQZ basis set from which it is derived, see Table 1. Half of this differences appears at the SCF level. We therefore repeat the MP2 and
CCSDŽT. calculations using our enhanced cc-pVX basis set. The increase in atomization between the cc-pV and cc-pVX basis sets is the largest for the TZ basis set, and decreases in magnitude as the basis set increases in size. The effect of enhancing the basis set is less than 0.1 kcalrmol on the 3-point extrapolated CCSDŽT. values. At the MP2 level, the ny4 q ny6 extrapolated value does not change when the basis set is improved from cc-pV to cc-pVX , while the MP2 value obtained using the variable a method actually decreases when the basis set is improved. Our experience w23x has been that the most reliable atomization energies are obtained for the basis set series with the most consistent results for the different extrapolation approaches. Thus we take our extrapolated value of 31.1 kcalrmol as our best estimate of the complete basis set MP2 result, which is similar to the 31.4 kcalrmol MP2-R12 value of Klopper and Almlof. ¨ Our best Žcc-pV5ZX . computed value of 30.7 kcalrmol is much better than the MP2 value Klopper and Almlof ¨ obtained with their big basis set. This again confirms the accuracy of the modern cc-pV basis sets. The relativistic effect on the atomization energy is y0.39 kcalrmol, see Table 1. This is small, but not insignificant. It is difficult to assign an error to this quantity, so we take a conservative approach and assume that it could be in error by "0.20 kcalrmol. We estimate our best atomization energy as follows. We take the ny4 q ny6 ŽTQ5. CCSDŽT. value from the cc-pVX enhanced basis set as our best value. We assume this extrapolated value is within about 0.5 kcalrmol of the CBS value on the basis of the difference between our extrapolated MP2 value using the enhanced basis set Ž31.1 kcalrmol. and the MP2-R12 value of Klopper and Almlof ¨ Ž31.4 kcalrmol.. Also the difference between our cc-pV5ZX value Ž26.67 kcalrmol. and the extrapolated value Ž26.97 kcalrmol. is only 0.3 kcalrmol, and the extrapolation should be accurate to within a factor of two. LRT found that the CCSDŽT. value was 0.2 kcalrmol smaller than the MRCI value in a small basis set that only recovered about half the binding energy. We therefore assume that the effect of higher levels of correlation could be twice that found by LRT, and thus correct our atomization energy by 0.4 " 0.4 kcalrmol for higher level correlation effects. We correct the atomization energy using the
C.W. Bauschlicher Jr., H. Partridger Chemical Physics Letters 300 (1999) 364–368
367
Table 2 Summary of extrapolated Mg 4 atomization energies, in kcalrmol Method
a
ny3 TQ
ny3 Q5
˚ cc-pV basis at B3LYPr6-311G ) re s 3.1719 A MP2 30.12 30.26 CCSDŽT. 26.86 26.67 cc-pV basis at CCSDŽT.rcc-pVTZ re s 3.1025 MP2 30.99 31.20 MP2–BSSE 30.76 CCSDŽT. 27.22 27.07 CCSDŽT. –BSSE 27.23 X cc-pV basis sets at re s 3.1025 MP2 31.32 31.23 CCSDŽT. 27.48 27.14 a b
ny4 TQ
ny4 Q5
ny4 q ny6 TQ5
a b TQ5
29.84 26.54
30.09 26.51
30.18 26.50
30.35Ž3.086. 26.49Ž4.104.
30.70 30.42 26.89 26.88
31.01
31.12
31.35Ž2.946.
26.90
26.91
26.91Ž3.949.
31.08 27.22
31.11 27.03
31.12 26.97
31.13Ž3.868. 26.94Ž4.884.
The extrapolation method is given in the first line and the basis sets used in the second. The alpha is given in parentheses.
CCSDŽT. core-valence effect of y0.94 kcalrmol. It is difficult to estimate the error in the core-valence contribution, but we assume that it can be as large as 50% of the computed effect. Thus our final De value is 26.97 " 0.5 y 0.94 " 0.47 q 0.4 " 0.4 s 26.4 " 1.4 kcalrmol, assuming that the error estimates are additive. This value is similar to that of Klopper and Almlof ¨ Ž27.8 kcalrmol., but smaller than the less rigorous estimate of LRT Ž31 kcalrmol.. We note that the Klopper and Almlof ¨ valence value is 27.6 kcalrmol, which is very similar to our valence estimate of 27.4 " 0.9 kcalrmol; the main difference between our best estimate and that of Klopper and Almlof ¨ is a result of the difference in their MP2 Ž0.2 kcalrmol. and our CCSDŽT. Žy0.94 kcalrmol. core effect. To obtain our final value, we include the zeropoint energy and scalar relativistic effects, which were not considered by Klopper and Almlof. ¨ Using our computed DK scalar relativistic effect and the CCSDŽT. zero-point energy w3x, yields our best estimate of the Mg 4 atomization energy of 24.6 " 1.6 kcalrmol.
4. Conclusions Our best MP2 value is in good agreement with the MP2-R12 value of Klopper and Almlof. ¨ Our extrapolated MP2 value is only 0.3 kcalrmol smaller than
the MP2-R12 value. The core-valence effect at the CCSDŽT. level has the opposite sign as determined at the MP2 level. Our best atomization energy, including zero-point energy and scalar relativistic effects, is 24.6 " 1.6 kcalrmol. References w1x T. Yamada, J. Vac. Sci. Technol. A. 16 Ž1998. 1403. w2x T. Yamada, C.W. Bauschlicher, H. Partridge, Phys. Rev. B., submitted. w3x T.J. Lee, A.P. Rendell, P.R. Taylor, J. Chem. Phys. 92 Ž1990. 489; 93 Ž1990. 6636. w4x R.J. Bartlett, Annu. Rev. Phys. Chem. 32 Ž1981. 359. w5x K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157 Ž1989. 479. w6x W. Klopper, J. Almlof, ¨ J. Chem. Phys. 99 Ž1993. 5167. w7x L.A. Curtiss, K. Raghavachari, G.W. Trucks, J.A. Pople, J. Chem. Phys. 94 Ž1991. 7221. w8x A. Ricca, C.W. Bauschlicher, J. Phys. Chem. 102 Ž1998. 876. w9x T.H. Dunning, J. Chem. Phys. 90 Ž1989. 1007. w10x R.A. Kendall, T.H. Dunning, R.J. Harrison, J. Chem. Phys. 96 Ž1992. 6796. w11x D.E. Woon, T.H. Dunning, J. Chem. Phys. 98 Ž1993. 1358. w12x D.E. Woon, T.H. Dunning Jr., to be published. w13x T. Helgaker, W. Klopper, H. Koch, J. Noga, J. Chem. Phys. 106 Ž1997. 9639. w14x J.M.L. Martin, Chem. Phys. Lett. 259 Ž1996. 669. w15x A.D. Becke, J. Chem. Phys. 98 Ž1993. 5648. w16x P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chem. 98 Ž1994. 11623. w17x M.J. Frisch, J.A. Pople, J.S. Binkley, J. Chem. Phys. 80 Ž1984. 3265, and references therein.
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w18x B.A. Hess, Phys. Rev. A 32 Ž1985. 756. w19x Gaussian 94, Revision D.1, M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, Gaussian, Inc., Pittsburgh PA, 1995. w20x MOLPRO is a package of ab initio programs written by H.-J. Werner, P.J. Knowles, with contributions from J. Almlof, ¨ R.D. Amos, A. Berning, D.L. Cooper, M.J.O. Deegan, A.J.
w21x
w22x w23x
Dobbyn, F. Eckert, S.T. Elbert, C. Hampel, R. Lindh, A.W. Llyod, W. Meyer, A. Nicklass, K. Peterson, R. Pitzer, A.J. Stone, P.R. Taylor, M.E. Mura, P. Pulay, M. Schutz, ¨ H. Stoll, T. Thorseinsson. The closed shell CCSD program is described in C. Hampel, K. Peterson, H.-J. Werner, Chem. Phys. Lett. 190 Ž1992. 1. MOLECULE-SWEDEN is an electronic structure program written by Almlof ¨ J, Bauschlicher CW, Blomberg MRA, ˚ RenChong DP, Heiberg A, Langhoff SR, Malmqvist P-A, dell AP, Roos BO, Siegbahn PEM, Taylor PR. TITAN is a set of electronic structure programs written by T.J. Lee, A.P. Rendell, J.E. Rice. C.W. Bauschlicher, A. Ricca, J. Phys. Chem. 102 Ž1998. 8044.