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Revised theoretical energy for the a 3Σg+ state of H2
REVISED
THEORETICAL
ENERGY
FOR THE a ‘LZ; STATE OF H2
w. KOLOS hIax Plamk Insfitrrte for PI:,ysics and Astrophysics, S Murk/l 40, Federal Republic of Gcrrnmy, oml Quarlturn Cl~+istry Group, Utu’versily of Warsaw, 02-093 Warsaw. PoZandf Received 31 October
1974
The Born-Oppenheimer energy of Hz in the a 3Zz state has been improved. The new result, with cstimatcd val?ICSof the relativistic and nonsdiabatjc corrections, azd wi’th :hc previously computed ndinbztic corrections yields To = 95077.3 cm ml in n good agreement with the rcccnt cxpcrimental value To = 95076.4 r 0.5 cm-’ of hlikr and Frcund.
The relative location of the triplet states with respect to the singlets in the hydrogen molecule has been investigated both expe,imentally and theoretically [l-4] . Recently the energ,’ difference between the u = 0,J = 0 levels of the a 3Ci and X ‘2’ states has been determined very accurately by Mi l?er nnd
in the previous computation [3,4] ) and tie energy has been cakulatcd for a few internuclear distances using an 80-term wavefunction which, due to a larger number of terms, was more flexible than the p&ious one. The results are listed in table I. wbcte the
Freund
give= in atomic
[5] who obtained
TO = 95076.4
f 0.5 cm-l.
This is in a fairly-good agreement wit11 the most accurate theoretical value To = 95079.8 cm-l [3,4], however, the residual small discrepancy is considerably larger than the experimental error. The theoretical value of To mentioned above has been obtained as the difference between the theoretical energy for the a 3Ef. state and that for the X lCf state. The latter is pro Eably v&y accurate, and therfiefore the upper state is more likely to be responsible for the existing discrepancy. The conceivable errors in the triplet state energy may be due to: (a) inaccurate Born-Oppenheimer Fotential energy curve, (b) neglect of the change of the relativistic and radiative. corrections due to the bond formation, and cornplete neglect of the radiative corrections for the eti cited electron, (c) inaccurate values of the diagonal corrections for nuclear motion,(d) neglect of the nonk!iabatic cprrections. To verify point (a) we have made an extensive calculation of the clamped nuclei energy in the a 3Ci state. h-larger basis has been used than that employed LL
I Permanent
addiess.
”
internuclear
R, nnd the total
distance, -anits
nnd
are
mentioning that fo: R = 1.87 even with a relatively simple 47-term wavefunction we obtained Do = 24614.68 &n -l. Thus the remaining 33 seleckd terms lowered the.energy by only 0.38 cm-l _By increasing the expansion to 100 terms an additional improvement of 0.05 cm-1 over the value listed in table l-was obtained. It is therefore not IikeIy that a still more extensive cdmputation would give e considerably better value of the energy. Since the basis set used in the present work did not contain terms with r312 a test of their importance has
Table 1 Theorerical energies for :he a 3~:c stzte of Hz compured Born-Oppenheimer appro:imakn 2) R
E
D
D -Do16
1.2 1.87 4.0 8.0
,-0.676680334 -0.737154489
11342.52 24615.06
c.4 0.6
-0.661565919.
8025.29
L.5
-0.626363653.
299.29
2.2
in the
D timcm-’ .
43
.‘. .:
cnzr~
energy, D, in
cm -1 . It is worth
a) R arid E in,atomic tiits,
:
bind&
tluz
..
:’
riumbcr 1‘ :’ CHEMICAL PHYSICS LI?l-TERS -’ -’ 15 F&wary 19,7j .‘.,: .bbn mide for th; ground stntc of PI, ..For R = 1.4, : ‘nuclear distance, rotational constant.and harmonic with 66,terms.selected .from the present basis set ,-we ‘.. vibrational frequency, respectively,..AE denotes-the .obthined the’binding energy De 7 38292.46 cm-1 ., “average” energy of excitation, to electronic states : W1;i~h.j~ by only 6.37 cm-t -worse than the lOO&rm of the same symmetry as the st,ate under consi.deraresult obtained.with a wavefunction depending or; tion, and O,,,r represents essentially part of the nu.?& ]6] . S’mce ir an excited triplet state the effect of clear mqtion corrections. the Coulomb correlation is certainly smaller than in In van Vleck’s notation one of the diagonal nuthe ground state: we may conclude that for the a 3Zi clear motion corrections [4] is state the ‘terms with r312 would give a practically neg. (H&,,, = (-ki A&= P, + Qnn, (3)‘. ligible’ contribution to the energy. : For larger internuclear distances the improvement with Qnr, related.to 0, .by Volmx~31, ,:.
oZ.Lhe
ehkrgy-obtained
in t!:e present
calculation
is
Q nn = Q2/2wjo,,:
seen to be somewhat lagger than forA =R, but this -hardly affects the lowest vibrational level. Hence,
Sin.de in both limiting have P, = 2Q,,, or
.from. the results iisted in tab!e 1, we may infer,that the present values of the clamped nuclei energy lower the zerothvibrationai level by about 0.7 cm-l. The overall improvement of the potential cnergy.curve was in our opinion too small to warrant its complete recomputation and a more accurate determination of the lowering of the lowest energy level. : cram ihe apprbximatious listed as point (b) the most serious one is the assumption that the relativistic corrections havethe same values for R.= R, as for R = m. A &h es t’m-ration of this error can easily be made. The a ?$ .state is the first one in a Rydherg series: Hence for the outer !oosely bcund electron we may assume a constant value of the relativistic corrections, the same as for a 2s electron, whereas for.tlie inner electron their value can be estimated from the results for the Hz ion [7] _Ir. this way for R = R, one gets a lowering of the total energy by. about 0.3 cm-l _ It is.npt possible,tq assess the accuracy of the di,agonal corrections for nuclear’motion mentioned above as point (c) which were calculated [4] .with a somewhat less accurate wavefunction than that used in the calculation:of the pptential energy curGe.‘However, a very rough es&nation of the nonadiabatic. ‘corrections can be made. ‘. ~&&ding to van’vleck [8] the nonadiabatic shift ‘of,thk zeroth vibrational level is ,. .A&~ W,6*/2,
-.
‘.
.:,
-. :
-
witli 6 d,_~_2&“,I~,
where R,, fle a,nd wc denote : -. _..,@‘._j, - ;:; .,
;,
‘,
‘.’ .’ -. :
the equilibrium .:
,inter-
(4) cases, R + 0 and R + m, we
4(fqnn = Q,,;
(5)
a very rough estimation of the nonadiabatic correction can be made by assuming eq. (5) to hold for all. R. ,.. From eqs. (l),.(2), (4) and (5), using for the a ?Zi state (Hl )nn -2 9 cm-l’ 7w e = 2700 cm-l, A.!?= 17000 cm-l, .which is the energy difference between the a and 113Zz states, one gets Aw = 1.5 cm’-‘. Thus the in;proved clamped nuclei energy, with the estimated relativistic and nonadiabatic corrections; gives for the a 3Z+ state To = 95077.3 cm-l which is very.close to tl? e new experimental value To = 95076.4 k 0.5 cm -I [5]. Since the estimated value of the nonadiabatic.correction represents probably nn upper limit the remaining small discrepancy seems to indicate that the relativistic and/or adiabatic corrections are subject to some improvement. The new,experimental value of To changes.the cxperlmental di%ociation energy in the a 3Zl .of H, which can be obtained as Do =IP H2 - Eii; + E,
.(‘I
‘.
.Ji)
-,
.,f E, 2, IS
T,, = 23300.5
cm-l, ‘. (6).
where IPI_12 denotes the ionization potential of HZ ‘: ‘and the other s$nbols are self-explanatory. The following numeri:al values have been.used in (6): IP,, = 124417 2 cm-I. [9] E pm~l-~lossm~ +-_10967;_8 crn-1 [lo] >EHI;= ’ H2c = -27419.7 cm-l which is the average energy for the j = l/2 and j = 3/2 states. When the previous theoretical dissocia-,. tion energy D,‘= 23297.4 cm-l [4] is adjusted to include the computed correction to the I3oru_Oppen-~ .I ‘. :’ : 8.. ., . . .’ .. .. ‘. : :,
Volume 3 I, number 1
CHEMICAL PHYSICS Ll3TERS
heimer potential, and the estimated relativistic and nonadiabatic corrections one gets D, = 23299.9 cm-l in a good agreement with the above experimental value. It is a great pleasure_to acknowledge the very kind ho$itality of Dr. G.H.F. Diercksen at the Max Planck Institute for Physics and Astrophysics in Munich.
References 111 H. Beutler and H.O. JErxer. 2. PhQsik IO1 (1936) 285.
[2] G.H. Dieke, J. Mol. Specrry. 2 (1958) 494. [ 31 W. Koios, Chem. Phys. Letters 1 (1967) 19. [4] W. KoTos and L. Wolnicwicz, 1. Chem. Phys. 48 (1968) 3672.
R.S. I-reund, J. Chcm. Phvs. 6: (1974) 2160. [6] W. Koios and L. Wolniewicz, J. Chcm. Phys. 49 (1968) 404. [7] SK. Luke, G. Hunter, R.P. ,SIcl%chran and hf. Cohen, J. Chem. Phys. 50 (1969) 1644. [8] J.H. van Vleck, J. Chem. Phys. 4 (1936) 327. .19 .1 C. Herzberg and C.H. Jun~en, J. hfol. Spectry. 4L (1972) 424. [lOI B. Jcziorski and W. KoTor, Chem. Phys. Letters 3 (1969) [5 ] T.A. hlillcr and
THEORETICAL
ENERGY
FOR THE a ‘LZ; STATE OF H2
w. KOLOS hIax Plamk Insfitrrte for PI:,ysics and Astrophysics, S Murk/l 40, Federal Republic of Gcrrnmy, oml Quarlturn Cl~+istry Group, Utu’versily of Warsaw, 02-093 Warsaw. PoZandf Received 31 October
1974
The Born-Oppenheimer energy of Hz in the a 3Zz state has been improved. The new result, with cstimatcd val?ICSof the relativistic and nonsdiabatjc corrections, azd wi’th :hc previously computed ndinbztic corrections yields To = 95077.3 cm ml in n good agreement with the rcccnt cxpcrimental value To = 95076.4 r 0.5 cm-’ of hlikr and Frcund.
The relative location of the triplet states with respect to the singlets in the hydrogen molecule has been investigated both expe,imentally and theoretically [l-4] . Recently the energ,’ difference between the u = 0,J = 0 levels of the a 3Ci and X ‘2’ states has been determined very accurately by Mi l?er nnd
in the previous computation [3,4] ) and tie energy has been cakulatcd for a few internuclear distances using an 80-term wavefunction which, due to a larger number of terms, was more flexible than the p&ious one. The results are listed in table I. wbcte the
Freund
give= in atomic
[5] who obtained
TO = 95076.4
f 0.5 cm-l.
This is in a fairly-good agreement wit11 the most accurate theoretical value To = 95079.8 cm-l [3,4], however, the residual small discrepancy is considerably larger than the experimental error. The theoretical value of To mentioned above has been obtained as the difference between the theoretical energy for the a 3Ef. state and that for the X lCf state. The latter is pro Eably v&y accurate, and therfiefore the upper state is more likely to be responsible for the existing discrepancy. The conceivable errors in the triplet state energy may be due to: (a) inaccurate Born-Oppenheimer Fotential energy curve, (b) neglect of the change of the relativistic and radiative. corrections due to the bond formation, and cornplete neglect of the radiative corrections for the eti cited electron, (c) inaccurate values of the diagonal corrections for nuclear motion,(d) neglect of the nonk!iabatic cprrections. To verify point (a) we have made an extensive calculation of the clamped nuclei energy in the a 3Ci state. h-larger basis has been used than that employed LL
I Permanent
addiess.
”
internuclear
R, nnd the total
distance, -anits
nnd
are
mentioning that fo: R = 1.87 even with a relatively simple 47-term wavefunction we obtained Do = 24614.68 &n -l. Thus the remaining 33 seleckd terms lowered the.energy by only 0.38 cm-l _By increasing the expansion to 100 terms an additional improvement of 0.05 cm-1 over the value listed in table l-was obtained. It is therefore not IikeIy that a still more extensive cdmputation would give e considerably better value of the energy. Since the basis set used in the present work did not contain terms with r312 a test of their importance has
Table 1 Theorerical energies for :he a 3~:c stzte of Hz compured Born-Oppenheimer appro:imakn 2) R
E
D
D -Do16
1.2 1.87 4.0 8.0
,-0.676680334 -0.737154489
11342.52 24615.06
c.4 0.6
-0.661565919.
8025.29
L.5
-0.626363653.
299.29
2.2
in the
D timcm-’ .
43
.‘. .:
cnzr~
energy, D, in
cm -1 . It is worth
a) R arid E in,atomic tiits,
:
bind&
tluz
..
:’
riumbcr 1‘ :’ CHEMICAL PHYSICS LI?l-TERS -’ -’ 15 F&wary 19,7j .‘.,: .bbn mide for th; ground stntc of PI, ..For R = 1.4, : ‘nuclear distance, rotational constant.and harmonic with 66,terms.selected .from the present basis set ,-we ‘.. vibrational frequency, respectively,..AE denotes-the .obthined the’binding energy De 7 38292.46 cm-1 ., “average” energy of excitation, to electronic states : W1;i~h.j~ by only 6.37 cm-t -worse than the lOO&rm of the same symmetry as the st,ate under consi.deraresult obtained.with a wavefunction depending or; tion, and O,,,r represents essentially part of the nu.?& ]6] . S’mce ir an excited triplet state the effect of clear mqtion corrections. the Coulomb correlation is certainly smaller than in In van Vleck’s notation one of the diagonal nuthe ground state: we may conclude that for the a 3Zi clear motion corrections [4] is state the ‘terms with r312 would give a practically neg. (H&,,, = (-ki A&= P, + Qnn, (3)‘. ligible’ contribution to the energy. : For larger internuclear distances the improvement with Qnr, related.to 0, .by Volmx~31, ,:.
oZ.Lhe
ehkrgy-obtained
in t!:e present
calculation
is
Q nn = Q2/2wjo,,:
seen to be somewhat lagger than forA =R, but this -hardly affects the lowest vibrational level. Hence,
Sin.de in both limiting have P, = 2Q,,, or
.from. the results iisted in tab!e 1, we may infer,that the present values of the clamped nuclei energy lower the zerothvibrationai level by about 0.7 cm-l. The overall improvement of the potential cnergy.curve was in our opinion too small to warrant its complete recomputation and a more accurate determination of the lowering of the lowest energy level. : cram ihe apprbximatious listed as point (b) the most serious one is the assumption that the relativistic corrections havethe same values for R.= R, as for R = m. A &h es t’m-ration of this error can easily be made. The a ?$ .state is the first one in a Rydherg series: Hence for the outer !oosely bcund electron we may assume a constant value of the relativistic corrections, the same as for a 2s electron, whereas for.tlie inner electron their value can be estimated from the results for the Hz ion [7] _Ir. this way for R = R, one gets a lowering of the total energy by. about 0.3 cm-l _ It is.npt possible,tq assess the accuracy of the di,agonal corrections for nuclear’motion mentioned above as point (c) which were calculated [4] .with a somewhat less accurate wavefunction than that used in the calculation:of the pptential energy curGe.‘However, a very rough es&nation of the nonadiabatic. ‘corrections can be made. ‘. ~&&ding to van’vleck [8] the nonadiabatic shift ‘of,thk zeroth vibrational level is ,. .A&~ W,6*/2,
-.
‘.
.:,
-. :
-
witli 6 d,_~_2&“,I~,
where R,, fle a,nd wc denote : -. _..,@‘._j, - ;:; .,
;,
‘,
‘.’ .’ -. :
the equilibrium .:
,inter-
(4) cases, R + 0 and R + m, we
4(fqnn = Q,,;
(5)
a very rough estimation of the nonadiabatic correction can be made by assuming eq. (5) to hold for all. R. ,.. From eqs. (l),.(2), (4) and (5), using for the a ?Zi state (Hl )nn -2 9 cm-l’ 7w e = 2700 cm-l, A.!?= 17000 cm-l, .which is the energy difference between the a and 113Zz states, one gets Aw = 1.5 cm’-‘. Thus the in;proved clamped nuclei energy, with the estimated relativistic and nonadiabatic corrections; gives for the a 3Z+ state To = 95077.3 cm-l which is very.close to tl? e new experimental value To = 95076.4 k 0.5 cm -I [5]. Since the estimated value of the nonadiabatic.correction represents probably nn upper limit the remaining small discrepancy seems to indicate that the relativistic and/or adiabatic corrections are subject to some improvement. The new,experimental value of To changes.the cxperlmental di%ociation energy in the a 3Zl .of H, which can be obtained as Do =IP H2 - Eii; + E,
.(‘I
‘.
.Ji)
-,
.,f E, 2, IS
T,, = 23300.5
cm-l, ‘. (6).
where IPI_12 denotes the ionization potential of HZ ‘: ‘and the other s$nbols are self-explanatory. The following numeri:al values have been.used in (6): IP,, = 124417 2 cm-I. [9] E pm~l-~lossm~ +-_10967;_8 crn-1 [lo] >EHI;= ’ H2c = -27419.7 cm-l which is the average energy for the j = l/2 and j = 3/2 states. When the previous theoretical dissocia-,. tion energy D,‘= 23297.4 cm-l [4] is adjusted to include the computed correction to the I3oru_Oppen-~ .I ‘. :’ : 8.. ., . . .’ .. .. ‘. : :,
Volume 3 I, number 1
CHEMICAL PHYSICS Ll3TERS
heimer potential, and the estimated relativistic and nonadiabatic corrections one gets D, = 23299.9 cm-l in a good agreement with the above experimental value. It is a great pleasure_to acknowledge the very kind ho$itality of Dr. G.H.F. Diercksen at the Max Planck Institute for Physics and Astrophysics in Munich.
References 111 H. Beutler and H.O. JErxer. 2. PhQsik IO1 (1936) 285.
[2] G.H. Dieke, J. Mol. Specrry. 2 (1958) 494. [ 31 W. Koios, Chem. Phys. Letters 1 (1967) 19. [4] W. KoTos and L. Wolnicwicz, 1. Chem. Phys. 48 (1968) 3672.
R.S. I-reund, J. Chcm. Phvs. 6: (1974) 2160. [6] W. Koios and L. Wolniewicz, J. Chcm. Phys. 49 (1968) 404. [7] SK. Luke, G. Hunter, R.P. ,SIcl%chran and hf. Cohen, J. Chem. Phys. 50 (1969) 1644. [8] J.H. van Vleck, J. Chem. Phys. 4 (1936) 327. .19 .1 C. Herzberg and C.H. Jun~en, J. hfol. Spectry. 4L (1972) 424. [lOI B. Jcziorski and W. KoTor, Chem. Phys. Letters 3 (1969) [5 ] T.A. hlillcr and