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Analysis of the ortho H2PARA H2 hyperfine dimer spectrum
Chemical Physics 63 (19811257-261 North-Holland Publishing Company
ANALYSIS M. WAAIJER,
OF THE ORTHO M. JACOBS
HrPARA
DIMER
SPECTRUM*
and J. REUSS
Fysisch Laborarorircm, Knrholieke Unicersireir, 625
Received
H2 NYPERFINE
ED
,Vij,negm,
The
.Nerheriands
21 July 1981
The molecular beam magnetic-resonance measurements on (H& of the Nijmegan group are re-analyzed using a recently developed method which takes properly into account the continuum states. As a result we propose modifications of the anisotropic part of the potential to obtain agreement with the differential cross section measurements of the GGttingen group as well as with the dimer spectroscopic results.
1. Introduction
2. MoIeruIar parameter
The magnetic hyperfine measurements of Verberne on Hz-Hz [l] will be re-examined in the following. We will restrict ourselves to the ortho Hz-para Hz system since Verberne’s fit of the experimental hyperfine spectrum is almost completely determined by the (L, J, F, F’) = (0, 1, 0, I) transition of ortho Hz-para Hz. As for HZ-H2 only the L = 0 and L = 1 states are bound, it is not correct to use the parameter 1VJ(E~-EO)~ (or 1V&B, as Verberne does) but one rather should work with c,. [V&(E2EJ]?, which is a sum over all continuum states
The frequency shift ilW of the (I., I, F, F’) = (0, 1, 0, 1) transition is determined by the mixing of L = 0 and L’ = 2 states. As L’ = 2 is not bound, the determining factor is a summation C,.f;,~,.O over continuum states Y’ (and L’ = 2) with
This system has found much attention of theoretical chemists, both due to its fundamental importance (simple system suitable for ab initio calculations) and due to astrophysical considerations (problems like cloud condensation and planetary atmospheres).
c1(2,0)
PI-
* This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)” and has been made possible by financial support from the “Nederlandse Stichting voor Zuiver Wetenschappelijk Onderzoek (ZWO)“.
0301-0104/81/0000-0000/$02.75
c1(2,0)-----
2
v-0
E,-Eo
l+c*(2,2)~-
VT0 VzO E2 - Eo
I
--1 --lil
1I
v.
(1)
and = -0.282843,
c2(2,2)
= so.199999. (2)
For the frequency
shift one obtains
A W(6, , 0, 1) = ;fi,,~.o(-S60.136)
kHz.
(3)
The symbols and quantum numbers are defined in ref. [2]. As / Vz,J(E1-E&
fk.”
@ 1981 North-Holland
=c:e,0,(&j;,.
Table I AWCY’) = AW(& 1.0, 1; v’) as a function of U’ with relevant quantities. calculated with the potential of Meyer [3]
With eqs. (4), (2) and (3) one obtains
1,*,1)=-44.81\7(~):.kHz. (5)
AW@,
I.’
Thus, the frequency shift AW(s, ortho HZ-para Hz is determined molecular parameter ; [ V?III(E? -
1, 0,l)for by one
0 I 2 3 J 5 6 7 8 9
E,,!lL
3. Analysis Verberne’s
experiment
AW&b,
2.267 10.157 13.576 11.029 9.696 8.952 8.334 7.695 6.988 6.204
E2-E,
(GHz)
[ VZ,,~‘Z~-
87.992 106.707 126.816 161.831 211.174 272.926 346.642
6.636x 9.060 x 1.146x 4.645 x 2.108 x 1.076 x 5.780x 3.170 x 1.743 x 9.446 x
432.165 529.420 638.335
-2
En)],.
1o-J IO+ lo-’ 1o-3 lo-’ lo-’ lo-’ 1o-J lo-” !O-’
1W(“‘) (Hz) -29 -411 -524 -210 -95 -48 -26 -14 -7 -4
[l] yields
N’,.,,(Ti, l,O, I)= 544.05(10) resulting in a frequency sector model value [2]
V2U (GHz)
kHz,
(6)
shift with respect to the
The results of eqs. (11) and (9) agree within 2% Summarizing, the potential of Meyer yields A W~fe,.rr (6, 1, 0, 1) = 1.37 kHz
l,O, I) =2.39(10)
kHz.
(.7)
Using eq. (5) one obtains for the experimental molecular parameter
(12)
and ‘I C [V~,l(E-E,)lt,]~~~s~~= Y.
0.0302(7).
(13)
These values differ strongly from the experimental values [eqs. (7) and (S)]. We compare this result with the one obtained from the HZ-HZ ab initio potential of Meyer [3], which is an adapted version of the potential used by Schafer and Meyer [4]. The frequency shift A-Wtfi, 1, 0, 1) calculated with this potentia1 of Meyer [3] consists of a sum over continuum states Y’ of A W(6, 1,0, 1; Y’), obtained by diagonalization of the energy matrix. Table 1 shows the contributions AW(v’) of the different states v’. Table 1 yields a total shift AW=~AW(v’)=1.369 Y’ and a molecular
kHz
(9)
parameter
Z[vZJ(E,--Eo)]t* Y.
=3.018x
lo-‘,
(10)
which with eq. (5) produces a shift A W = 1.352 kHz.
(11)
4. Discussion Buck et al. [S] state that the repu!sive part of Meyer has to be shifted O-l.& to smaller R values in order to be consistent -with their measurements of the differential cross sections of the O+ 1 rotational transition for HD f Dz [S]. This conclusion is independently arrived at by Silvera and Goldmann [6], who constructed a potential relying much on solid state data for the repulsive branch. Furthermore, according to Buck et al. [7], the ratio VT/V0 has to be retained in the repuisive region in order to obtain a reasonable fit to their differential cross-section measurements of the 0-t 2 rotational transition for H,+D,. Therefore, we tested a potential in which the repulsive part of the isotropic as weli as the
of the isotropic potential
Table 2 Long range coefficients II? 35.52 A) for Me~er’s and the present one [eqs. (13) and (15)]
anisotropic potential is shifted 0.1 A to smaller R values as compared to the potential of Meyer [3]. Since this was not su%cient to reproduce the experimental values [eqs. (6), (7) and (S)], we also added a correction term (blister) to the anisotropic potential in order to deepen the potential between the repulsive and the long range branch. As the potentiai of Meyer is partly given numerically, we fitted a series expansion to get an analytic expression. For the present shifted potential we used the same procedure. The potential of Meyer and the present potential are separated into two parts
n
C,,, (GHz A”)
C2,, (GHz A”,
5 6 7 8 10
+7.310 -7.9858X -1.462x
IO7 IO’
-1.4621 -1.6741 X lo6 ‘1.7476~ 10h
-1.4563x 199 -3.3153 x IO’”
-8.0114X 10’ -2.0215 X 10”
The coefficients Cc, and Cz, for 2.00 As R s 5.82 A [eqs. (16) and (17)] are the results of the potential fits. For the blister term V,, we assume
(14)
Vc2 = -75.459
(13 for R 35.82 Vo=
V, =
:
“=”
(16)
Cz,/ R” f Vc2,
(17)
for 2.00 As R s 5.82 A. The coefficients Co, and Czn for R z=S.82 A [eqs. (14) and (IS)] are those given by Meyer [3] converted from atomic units to GHz A”.
Table 3 Fitted Co, and Cz, n
0 1 2 3 4 5 6 7 8 9 10
coefficientsfor
exp I-14.4@
-3.30)‘]
GE,
AW=x
?,W(~‘)=2.416 1.’
kHz
Meyer’s potential and the present potential [rqs. (16) and (17)] for 2.00 .
Present potential
(18)
with R in A; it only applies to the present potential; to the potential of Meyer applies eq. (17) with Vcz = 0. Tables 2 and 3 tabulate all coefficients used. Fig. 1 shows a graphical comparison of Meyer’s potential with the present one. For the latter potential we repeat the calculations done with Meyer’s potential in the beginning of this section. Table 4 shows the results, yielding 2 total shift
A, and
f C&/R”, n=O
potenrial
Meyer’s
potential
Co, (GHz A”)
Cz, (GHz x”,
&,
-1822291 66778907 -1082026728 10201977312 -61951010569 253038254052 -703695004416 1315005034743 -1579551807085 1000%1106566 -338121387905
-892945 31807586 -501631472 4610355964 -27334489491 109195446329 -207559570873 545890403090 -64498714S138 443043193985 -134321981493
-3217571 116838403 -1882145112 17707419304 -107715284875 442535323041 -1243015888402 2355799673128 -2881409506097 20529014252u -646779869068
(GHz A”
I
Cz,, (GHz x” ) -1652510 60605736 -967869325 9021910852 -54345378208 220991471689 -614208906260 1131748509237 -1394070470096 993242293057 -306796072537
119)
Hz-para H2 hyperfine dimer spectrum
&f. Waaijer er al. I Or&
260 potential energy IlO* GHz)
Table 5 Comparison
12
of the potentia!s
Source
~[VZ,/(Ez-E,)];. I.
AW(kHz)
Meyer‘s potential present potential Verberne’s experiment
0.0302 (7) 0.0530 (11) 0.0.533 (23)
1.37 2.42 2.39 (IO)
oH2 -PHI
a
-
a
_---
b
with the experiment
c
and a molecular c
parameter
5 [Vzo/(E,-ECJ]Zy, In parentheses,
-L
-12
-16
i
z
3
L
Fig. 1. The present potential Meyer f b).
6
5
(a) compared
dl
to that of
Table 4 .lWiv’) = AWiG, LO, 1; v’) as a function of Y’ with quantities, calculated with the present potential 1.’
VZU (GHz)
Ez -En (GHr)
[ VXJ Ez- E,)lt.
0 1
4.482 18.895 17.848 13.831 12.632 12.072 11.614 11.108
109.155 124.707 143.661 180.063 229.630
1.686X 10-j 2.296 x lo-’ 1.544 X lo-’ 5.900 x10-j 3.026 x 1O-3
291.216 364.598 449.685 546.411
1.718X 1.015 x 6.102 x 3.7lxx 2.241 x
2 3 4 5 6 7 8 9
10.507 9.800
654.719
10-j 1o-3 lo-” 10-d lo-”
relevant
AlV(v’) 0-W -75 -1057 -706 -267 -136 -77 -45 -27 -16 -10
(20)
with eq. (5) one obtains a shift
AW = 2.373 kHz, -8
= 5.295 x lo-‘.
(21)
in agreement with eq. (19) within 2%. Table 5 combines all results. The present potential fits the experiment of Verberne very wel!. The expectation is that also Buck’s experiment should be described rather well, because we incorporated all elements (the 0.1 A shift of the repulsive branch and its conserved ratio V2/ V,) which emerged as being of dominant influence [7]. Our present potential, however, should rather stimulate future theoretical work and experimental effort than be regarded as a final answer, for the blister term was introduced ad hoc to yield the proper molecular parameter [20]. After completion of this work we received a preprint by P.G. Burton and U.E. Senf of the University of Wollongong, Australia. The authors present improved ab initio potentials for (H&. Some of the shortcomings of the potential of Meyer [S] are remedied; however, a good agreement with experimental results is still lacking, probably due to intrinsic limitations of this kind of calculations.
References [I] J.F.C. Verberne, Thesis, Katholieke Universiteit, Nijmegen, The Netherlands (1979); J.Verbeme andJ.Reuss.Chem.Phys.50 (1980)137; 54 (1981) 189.
[2] M. Waaijer, M. Jacobs and J. Reuss, Chem. Phys. 63 (19811247. [33 W. Meyer’s adapted version, ccmmunicated to us by SchZfer (1980!. [4] J. SchZfer and W. Meyer, J. Chem. Phys. 70 (1979) 344.
[S] U. Buck, F. H&ken, J. Schleusener and J. SchZfer, J. Chem. Phys. 74 (1981) 535. [6] I.F. Silvera and V.V. Goldmann. J. Chem. Phys. 69 (1978) 4209. 171 U. Buck, private communication (1981).
ANALYSIS M. WAAIJER,
OF THE ORTHO M. JACOBS
HrPARA
DIMER
SPECTRUM*
and J. REUSS
Fysisch Laborarorircm, Knrholieke Unicersireir, 625
Received
H2 NYPERFINE
ED
,Vij,negm,
The
.Nerheriands
21 July 1981
The molecular beam magnetic-resonance measurements on (H& of the Nijmegan group are re-analyzed using a recently developed method which takes properly into account the continuum states. As a result we propose modifications of the anisotropic part of the potential to obtain agreement with the differential cross section measurements of the GGttingen group as well as with the dimer spectroscopic results.
1. Introduction
2. MoIeruIar parameter
The magnetic hyperfine measurements of Verberne on Hz-Hz [l] will be re-examined in the following. We will restrict ourselves to the ortho Hz-para Hz system since Verberne’s fit of the experimental hyperfine spectrum is almost completely determined by the (L, J, F, F’) = (0, 1, 0, I) transition of ortho Hz-para Hz. As for HZ-H2 only the L = 0 and L = 1 states are bound, it is not correct to use the parameter 1VJ(E~-EO)~ (or 1V&B, as Verberne does) but one rather should work with c,. [V&(E2EJ]?, which is a sum over all continuum states
The frequency shift ilW of the (I., I, F, F’) = (0, 1, 0, 1) transition is determined by the mixing of L = 0 and L’ = 2 states. As L’ = 2 is not bound, the determining factor is a summation C,.f;,~,.O over continuum states Y’ (and L’ = 2) with
This system has found much attention of theoretical chemists, both due to its fundamental importance (simple system suitable for ab initio calculations) and due to astrophysical considerations (problems like cloud condensation and planetary atmospheres).
c1(2,0)
PI-
* This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)” and has been made possible by financial support from the “Nederlandse Stichting voor Zuiver Wetenschappelijk Onderzoek (ZWO)“.
0301-0104/81/0000-0000/$02.75
c1(2,0)-----
2
v-0
E,-Eo
l+c*(2,2)~-
VT0 VzO E2 - Eo
I
--1 --lil
1I
v.
(1)
and = -0.282843,
c2(2,2)
= so.199999. (2)
For the frequency
shift one obtains
A W(6, , 0, 1) = ;fi,,~.o(-S60.136)
kHz.
(3)
The symbols and quantum numbers are defined in ref. [2]. As / Vz,J(E1-E&
fk.”
@ 1981 North-Holland
=c:e,0,(&j;,.
Table I AWCY’) = AW(& 1.0, 1; v’) as a function of U’ with relevant quantities. calculated with the potential of Meyer [3]
With eqs. (4), (2) and (3) one obtains
1,*,1)=-44.81\7(~):.kHz. (5)
AW@,
I.’
Thus, the frequency shift AW(s, ortho HZ-para Hz is determined molecular parameter ; [ V?III(E? -
1, 0,l)for by one
0 I 2 3 J 5 6 7 8 9
E,,!lL
3. Analysis Verberne’s
experiment
AW&b,
2.267 10.157 13.576 11.029 9.696 8.952 8.334 7.695 6.988 6.204
E2-E,
(GHz)
[ VZ,,~‘Z~-
87.992 106.707 126.816 161.831 211.174 272.926 346.642
6.636x 9.060 x 1.146x 4.645 x 2.108 x 1.076 x 5.780x 3.170 x 1.743 x 9.446 x
432.165 529.420 638.335
-2
En)],.
1o-J IO+ lo-’ 1o-3 lo-’ lo-’ lo-’ 1o-J lo-” !O-’
1W(“‘) (Hz) -29 -411 -524 -210 -95 -48 -26 -14 -7 -4
[l] yields
N’,.,,(Ti, l,O, I)= 544.05(10) resulting in a frequency sector model value [2]
V2U (GHz)
kHz,
(6)
shift with respect to the
The results of eqs. (11) and (9) agree within 2% Summarizing, the potential of Meyer yields A W~fe,.rr (6, 1, 0, 1) = 1.37 kHz
l,O, I) =2.39(10)
kHz.
(.7)
Using eq. (5) one obtains for the experimental molecular parameter
(12)
and ‘I C [V~,l(E-E,)lt,]~~~s~~= Y.
0.0302(7).
(13)
These values differ strongly from the experimental values [eqs. (7) and (S)]. We compare this result with the one obtained from the HZ-HZ ab initio potential of Meyer [3], which is an adapted version of the potential used by Schafer and Meyer [4]. The frequency shift A-Wtfi, 1, 0, 1) calculated with this potentia1 of Meyer [3] consists of a sum over continuum states Y’ of A W(6, 1,0, 1; Y’), obtained by diagonalization of the energy matrix. Table 1 shows the contributions AW(v’) of the different states v’. Table 1 yields a total shift AW=~AW(v’)=1.369 Y’ and a molecular
kHz
(9)
parameter
Z[vZJ(E,--Eo)]t* Y.
=3.018x
lo-‘,
(10)
which with eq. (5) produces a shift A W = 1.352 kHz.
(11)
4. Discussion Buck et al. [S] state that the repu!sive part of Meyer has to be shifted O-l.& to smaller R values in order to be consistent -with their measurements of the differential cross sections of the O+ 1 rotational transition for HD f Dz [S]. This conclusion is independently arrived at by Silvera and Goldmann [6], who constructed a potential relying much on solid state data for the repulsive branch. Furthermore, according to Buck et al. [7], the ratio VT/V0 has to be retained in the repuisive region in order to obtain a reasonable fit to their differential cross-section measurements of the 0-t 2 rotational transition for H,+D,. Therefore, we tested a potential in which the repulsive part of the isotropic as weli as the
of the isotropic potential
Table 2 Long range coefficients II? 35.52 A) for Me~er’s and the present one [eqs. (13) and (15)]
anisotropic potential is shifted 0.1 A to smaller R values as compared to the potential of Meyer [3]. Since this was not su%cient to reproduce the experimental values [eqs. (6), (7) and (S)], we also added a correction term (blister) to the anisotropic potential in order to deepen the potential between the repulsive and the long range branch. As the potentiai of Meyer is partly given numerically, we fitted a series expansion to get an analytic expression. For the present shifted potential we used the same procedure. The potential of Meyer and the present potential are separated into two parts
n
C,,, (GHz A”)
C2,, (GHz A”,
5 6 7 8 10
+7.310 -7.9858X -1.462x
IO7 IO’
-1.4621 -1.6741 X lo6 ‘1.7476~ 10h
-1.4563x 199 -3.3153 x IO’”
-8.0114X 10’ -2.0215 X 10”
The coefficients Cc, and Cz, for 2.00 As R s 5.82 A [eqs. (16) and (17)] are the results of the potential fits. For the blister term V,, we assume
(14)
Vc2 = -75.459
(13 for R 35.82 Vo=
V, =
:
“=”
(16)
Cz,/ R” f Vc2,
(17)
for 2.00 As R s 5.82 A. The coefficients Co, and Czn for R z=S.82 A [eqs. (14) and (IS)] are those given by Meyer [3] converted from atomic units to GHz A”.
Table 3 Fitted Co, and Cz, n
0 1 2 3 4 5 6 7 8 9 10
coefficientsfor
exp I-14.4@
-3.30)‘]
GE,
AW=x
?,W(~‘)=2.416 1.’
kHz
Meyer’s potential and the present potential [rqs. (16) and (17)] for 2.00 .
Present potential
(18)
with R in A; it only applies to the present potential; to the potential of Meyer applies eq. (17) with Vcz = 0. Tables 2 and 3 tabulate all coefficients used. Fig. 1 shows a graphical comparison of Meyer’s potential with the present one. For the latter potential we repeat the calculations done with Meyer’s potential in the beginning of this section. Table 4 shows the results, yielding 2 total shift
A, and
f C&/R”, n=O
potenrial
Meyer’s
potential
Co, (GHz A”)
Cz, (GHz x”,
&,
-1822291 66778907 -1082026728 10201977312 -61951010569 253038254052 -703695004416 1315005034743 -1579551807085 1000%1106566 -338121387905
-892945 31807586 -501631472 4610355964 -27334489491 109195446329 -207559570873 545890403090 -64498714S138 443043193985 -134321981493
-3217571 116838403 -1882145112 17707419304 -107715284875 442535323041 -1243015888402 2355799673128 -2881409506097 20529014252u -646779869068
(GHz A”
I
Cz,, (GHz x” ) -1652510 60605736 -967869325 9021910852 -54345378208 220991471689 -614208906260 1131748509237 -1394070470096 993242293057 -306796072537
119)
Hz-para H2 hyperfine dimer spectrum
&f. Waaijer er al. I Or&
260 potential energy IlO* GHz)
Table 5 Comparison
12
of the potentia!s
Source
~[VZ,/(Ez-E,)];. I.
AW(kHz)
Meyer‘s potential present potential Verberne’s experiment
0.0302 (7) 0.0530 (11) 0.0.533 (23)
1.37 2.42 2.39 (IO)
oH2 -PHI
a
-
a
_---
b
with the experiment
c
and a molecular c
parameter
5 [Vzo/(E,-ECJ]Zy, In parentheses,
-L
-12
-16
i
z
3
L
Fig. 1. The present potential Meyer f b).
6
5
(a) compared
dl
to that of
Table 4 .lWiv’) = AWiG, LO, 1; v’) as a function of Y’ with quantities, calculated with the present potential 1.’
VZU (GHz)
Ez -En (GHr)
[ VXJ Ez- E,)lt.
0 1
4.482 18.895 17.848 13.831 12.632 12.072 11.614 11.108
109.155 124.707 143.661 180.063 229.630
1.686X 10-j 2.296 x lo-’ 1.544 X lo-’ 5.900 x10-j 3.026 x 1O-3
291.216 364.598 449.685 546.411
1.718X 1.015 x 6.102 x 3.7lxx 2.241 x
2 3 4 5 6 7 8 9
10.507 9.800
654.719
10-j 1o-3 lo-” 10-d lo-”
relevant
AlV(v’) 0-W -75 -1057 -706 -267 -136 -77 -45 -27 -16 -10
(20)
with eq. (5) one obtains a shift
AW = 2.373 kHz, -8
= 5.295 x lo-‘.
(21)
in agreement with eq. (19) within 2%. Table 5 combines all results. The present potential fits the experiment of Verberne very wel!. The expectation is that also Buck’s experiment should be described rather well, because we incorporated all elements (the 0.1 A shift of the repulsive branch and its conserved ratio V2/ V,) which emerged as being of dominant influence [7]. Our present potential, however, should rather stimulate future theoretical work and experimental effort than be regarded as a final answer, for the blister term was introduced ad hoc to yield the proper molecular parameter [20]. After completion of this work we received a preprint by P.G. Burton and U.E. Senf of the University of Wollongong, Australia. The authors present improved ab initio potentials for (H&. Some of the shortcomings of the potential of Meyer [S] are remedied; however, a good agreement with experimental results is still lacking, probably due to intrinsic limitations of this kind of calculations.
References [I] J.F.C. Verberne, Thesis, Katholieke Universiteit, Nijmegen, The Netherlands (1979); J.Verbeme andJ.Reuss.Chem.Phys.50 (1980)137; 54 (1981) 189.
[2] M. Waaijer, M. Jacobs and J. Reuss, Chem. Phys. 63 (19811247. [33 W. Meyer’s adapted version, ccmmunicated to us by SchZfer (1980!. [4] J. SchZfer and W. Meyer, J. Chem. Phys. 70 (1979) 344.
[S] U. Buck, F. H&ken, J. Schleusener and J. SchZfer, J. Chem. Phys. 74 (1981) 535. [6] I.F. Silvera and V.V. Goldmann. J. Chem. Phys. 69 (1978) 4209. 171 U. Buck, private communication (1981).