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Partition functions and equilibrium constants for LiH and LiH+
J. Quant. Sptwrosc.
Radiat. Transfer Vol. 55, No. 6,
pp.849-852,
1996
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
Pergamon !300224073(%)OOO5G8
OO22-4073/96 $15.00 + 0.00
NOTE PARTITION FUNCTIONS CONSTANTS FOR
AND EQUILIBRIUM LiH AND LiH+
P. C. STANCIL W. M. Keck Laboratory for Computational Physics, Department of Physics, University of Nevada Las Vegas, Las Vegas, NV 89154-5002, U.S.A. (Received
24 October
1995)
Abstract-Partition functions and equilibrium constants for the diatomic molecules LiH and LiH+ are presented for temperatures Tup to 40,000 K. The results are obtained by a direct sum of all electronic ground state rotational-vibrational levels whose energies are determined by solving the radial nuclear Schrijdinger equation within the Born-Oppenheimer approximation. Copyright 0 1996 Elsevier Science Ltd
INTRODUCTION Lithium hydride has become a molecule of great interest in studies of the postrecombination era of the early universe. Because of LiH’s large dipole moment, it has been the subject of searches for rotational-vibrational (RV) lines at high redshifts in primordial clouds’ and has been postulated to partially erase spatial anisotropies in the cosmic background radiation (CBR) field by Thomson scattering of CBR photons.* Signore et al3 have suggested that if a large fraction of lithium is contained in LiH, measurements of the LiH RV lines could place constraints on the primordial lithium abundance and discriminate between various big bang nucleosynthesis models. In an analysis of the lithium chemistry of the postrecombination epoch, Stancil et al4 found the abundances of LiH, as well as it’s molecular ion LiH+ , to be so small as to cast doubt on the utility of the above proposals. It may still be possible to observe LiH lines, though, in gravitationally collapsing pre-galactic clouds since the LiH abundance is expected to dramatically increase. During the investigation reported in Ref. 4, discrepancies between the LiH partition functions obtained quantum-mechanically and the polynomial expansions of Sauval and Taturn’ were found. A similar situation for H: and He: was previously pointed out by Stancil.6 New partition functions and dissociation equilibrium constants for LiH and LiH+ and ionization equilibrium constants for LiH are presented.
THEORY
AND
CALCULATIONS
The molecular partition function is given by
& CT) = 1 L’
cN exp[- cEggN
J%N)/h T 1
(1)
for the molecule xH, where x = Li or Li +. The vibrational and rotational quantum numbers are given by u and N, respectively, EcNis the binding energy of the UN RV level, Eg is the binding energy of the lowest RV level, k,, is the Boltzmann constant, and g, is the total statistical weight given by’ 849
Note
850
Table 1. Maximum rotational quantum number N for each vibrational quantum number u of the ground state used in the partition function calculations. Molecule 0
1
2
3
4
V” 5
6
7
8
9
10
11
LiH
57
58
57
56
54
52
50
48
47
45
43
41
LiH+
16
13
11
8
6
4
-
-
-
-
-
-
2,
LiH LiH+
Table
2.
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 252 300 336 400 420 500 504 600 630 700 800 840 900 1000 1008 1100 1200 1260 1300 1400 1500 1600
14
15
16
17
18
19
20
21
22
34
32
30
27
25
22
20
18
15
"KDI
(cm 5
13 36
_
_
_
_
_
_
_
_
Partition functions Q and dissociation “K,, (Dl: LiH-+Li + H) and ionization LiH -+ LiH+ + e-) equilibrium constants for LiH as functions of temperature.
Q 10
12 39 _
1.0431+00 1.3682+00 2.2587+00 3.1881-tOO 4.1254+00 5.0658+00 6.0081-tOO 6.9515+00 7.8957$00 8.8407+00 9.7863+00 1.0732+01 1.1679+01 1.2626+01 1.3574+01 1.4522+01 1.5471+01 1.6420+01 1.7370+01 1.8320+01 1.9271+01 2.4230+01 2.8842+01 3.2339+01 3.8674+01 4.0693+01 4.9010+01 4.9437+01 6.0056+-01 6.3531+01 7.1955+01 8.4803+01 9.0223+01 9.8666+01 1.1359+02 1.1484+02 1.2963+02 1.4680+02 1.5766+02 1.6514+02 1.8468+02 2.0544+02 2.2745+02
-3
Q
"KII
1
1.6175-251 2.7978-204 1.2417-170 1.7613-145 5.4689-126 1.8751-110 8.6112-98 2.7954-87 2.0276-78 7.4714-71 2.5527-64 1.2691-58 1.2952-53 3.5449-49 3.2121-45 1.1401-41 7.1959-28 1.7291-19 8.0772-15 1.1915-08 4.0479-07 2.9829-02 4.7462-02 4.9081+02 4.8824+03 4.7516+05 7.9795+07 4.3748+08 4.2oa9+09 9.9231+10 1.2431+11 1.3065+12 1.1132+13 3.4124+13 6.7937$13 3.1918+14 1.2168+15 3.9153+15
(cm-S)
4.1714-280 9.9453-259 3.3798-240 5.5926-224 1.1528-209 6.1462-197 1.5091-185 2.7067-175 4.6102-135 2.8330-110 2.5635-96 1.0822-77 4.9414-73 4.1033-58 1.7181-57 4.7651-45 6.1670-42 1.0359-35 1.0475-28 2.2661-26 2.9428-23 6.7184-19 1.3758-18 2.4695-15 2.3059-12 8.2838-11 7.5022-10 1.0668-07 7.8031-06 3.3334-04
1680 1700 1800 1900 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000 21000 22000 23000 24000 25000 26000 27000 28000 29000 30000 31000 32000 33000 34000 35000 36000 37000 38000 39000 40000
2.459a+o2 2.5074+02 2.7534+02 3.0127+02 3.2856+02 6.8259+02 1.1986+03 1.8763+03 2.6898+03 3.6013+03 4.5737+03 5.5759+03 6.5844+03 7.5825+03 8.5584+03 9.5049+03 1.0417+04 1.1293$04 1.2131+04 1.2932+04 1.3696+04 1.4425+04 1.5119+04 1.5781+04 1.6411+04 1.7012+04 1.75a5+04 1.8132+04 1.8654+04 1.9153+04 1.9629+04 2.0085+04 2.0522+04 2.0940+04 2.1340+04 2.1724+04 2.2093$04 2.2447+04 2.2788+04 2.3115+04 2.3430+04 2.3733+04 2.4026+04
"KDI
(cm
-3
“K,,
(II:
“Kll
)
9.0086+15 1.0958+16 2.7303+16 6.1691+16 1.2827+17 1.2480+19 1.1571+20 4.3588+20 1.0901+21 2.2274+21 4.1078+21 ‘7.2016+21 1.2352+22 2.1074+22 3.6118-t22 6.2552+22 1.0985+23 1.9603+23 3.5579+23 6.5713+23 1.2350+24 2.3616-t24 4.5923+24 9.0773+24 1.8227+25 3.7161+25 7.6871+25 1.6125+26 3.4278+26 7.3802+26 1.6084+27 3.5463+27 7.9062+27 1.7813+28 4.0541+28 9.3160+28 2.1605+29 5.0543+29 1.1924+30 2.8355+30 6.7942+30 1.6399+31 3.9857+31
(cm-7
4.8641-03 9.1374-03
1.7305-01 2.4006+00 2.5558+01 7.7195+07 1.2527+11 1.0213+13 1.8997+14 1.5349+15 7.4056+15 2.5406+16 6.8726+16 1.5648+17 3.1310+17 5.6712+17 9.4983$17 1.4938+18 2.2318+18 3.1958-tl8 4.4163+18 5.9219+1s 7.7388$-18 9.8912+18 1.2400+19 1.5286+19 1.8564+19 2.2251+19 2.6358+19 3.0896+19 3.5876+19 4.1304+19 4.7188+19 5.3531+19 6.0339+19 6.7614-1-19 7.5359+19 8.3574+19 9.2260+19 1.0142+20 1.1105+20 1.2115+20 1.3171+20
Note Table 3. Partition
functions Q and dissociation equilibrium constants “Ko2 (LiH+ --t Li+ + H) and “Ko3 (LiH+ -+ Li + H+ ) for LiH+ as functions of temperature.
Q 5 10
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 252 300 336 400 420 500 504 600 630 700 BOO 840 900 1000 1008 1100 1200 1260 1300 1400 1500 1600
851
1.3556+00 2.2310+00 4.0722+00 5.9335+00 7.8034+00 9.6804+00 1.1566+01 1.3462+01 1.5378+01 1.7319+01 1.9297+01 2.1321+01 2.3401+01 2.5546+01 2.7767+01 3.0069+01 3.24601-01 3.4945+01 3.7528+01 4.0210+01 4.2995+01 5.9060+01 7.5948+01 8.9572+01 1.1500+02 1.2311+02 1.5557+02 1.5718+02 1.9481+02 2.0611+02 2.3143+02 2.6499+02 2.7755+02 2.9549+02 3.2312+02 3.2521+02 3.4814+02 3.7082+02 3.8342+02 3.9143+02 4.1021+02 4.2737+02 4.4309+02
"KDZ
"KDS
(cmeS)
( cmmS)
Q
5.5155-158
_ _
9.2128-71
3.4684-25 1.6390-09 1.4610-01 9.3099+03 1.5221+07 3.0202+09 1.5929+11 3.4615+12 4.0372+13 2.9919+14 1.5772+15 6.3960+15 2.1106+16 5.9052+16 1.4450+17 3.1669+17 6.3320+17 1.1721+18 2.0323+18 1.6858+19 5.9550+19 1.1938+20 2.9772+20 3.7352+20 7.7185+20 7.9564+20 1.4691+21 1.7173+21 2.3598+21 3.4172+21 3.8820+21 4.6207+21 5.9551+21 6.0672+21 7.4091+21 8.9742+21 9.9636+21 1.0643+22 1.2412+22 1.4274+22 1.6227+22
_ _ _ _ _ _ _ _ 4.2706-284 1.9678-263 2.2652-245 1.8822-229 2.5758-215 1.1132-202 2.5568-191 6.8423-147 1.1684-119 2.2076-104 4.5897-84 5.3954-79 8.5635-63 4.0450-62 1.2056-48 2.7991-45 1.5013-38 5.5706-31 1.8597-28 4.3151-25 2.2363-20 4.8580-20 1.6242-16 2.7038-13 1.3216-11 1.4497-10 3.1910-08 3.4450-06 2.0842-04
1680 1700 1800 1900 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000 21000 22000 23000 24000 25000 26000 27000 28000 29000 30000 31000 32000 33000 34000 35000 36000 37000 38000 39000 40000
4.5475+02 4.5754+02 4.7086+02 4.8316+02 4.9456+02 5.7455+02 6.2014+02 6.4950+02 6.6996+02 6.8503+02 6.9659+02 7.0573+02 7.1314+02 7.1927+02 7.2443+02 7.2882+02 7.3262+02 7.3592+02 7.3882+02 7.4140+02 7.4370+02 7.4576+02 7.4762+02 7.4931+02 7.5085+02 7.5225+02 7.5355+02 7.5474+02 7.5584+02 7.5686+02 7.5781+02 7.5870+02 7.5953+02 7.6030+02 7.6103+02 7.6172+02 7.6236+02 7.6297+02 7.6355+02 7.6409+02 7.6461+02 7.6510+02 7.6556+02
"KDZ
"KD~
( cmw3)
(cm )
1.7852i22 1.8267+22 2.0391+22 2.2597$22 2.4882+22 5.1646+22 8.4482$22 1.2232+23 1.6448+23 2.1051+23 2.6017+23 3.1334+23 3.7000+23 4.3020+23 4.9405+23 5.6172+23 6.3340+23 7.0933+23 7.8977$23 8.7500+23 9.6534+23 1.0611+24 1.1628+24 1.2706+24 1.3851+24 1.5068+24 1.6359+24 1.7732$24 1.9192+24 2.0743+24 2.2393+24 2.4148+24 2.6015+24 2.8001+24 3.0115+24 3.2365+24 3.4759+24 3.7308+24 4.0020+24 4.2908+24 4.5983-i-24 4.9256$24 5.2742+24
-3
3.9206-03 7.8242-03 1.9730-01 3.5578+00 4.8233+01 8.0511+08 3.7576+12 6.6070+14 2.2791+16 3.1431+17 2.4768+18 1.3596+19 5.8519+19 2.1280+20 6.8664+20 2.0328+21 5.6522+21 1.5009+22 3.8532$22 9.6494+22 2.3732+23 5.7619+23 1.3863+24 3.3155124 7.8997+24 1.8787+25 4.4654$25 1.0620+26 2.5292+26 6.0357+26 1.4440+27 3.4649+27 8.3402+27 2.0143128 4.8821+28 1.1875+29 2.8991+29 7.1033+29 1.7468+30 4.3111+30 1.0678+31 2.6539+31 6.6188+31
g, = (2N + 1).
(2)
The dissociation equilibrium constant in terms of number densities can be expressed by “K(T) =
n (x)n (I-0 n (xW -exp(-Di/k,T)
where n(x) is the number density in cme3, QXand QH are the partition functions of the atoms x and H, p is the molecular reduced mass, and 0: is the molecular dissociation energy. A similar expression can be given for the ionization equilibrium constant.
852
Note
To obtain the energies of the RV states, the radial nuclear Schriidinger equation was numerically solved using nonrelativistic Born-Oppenheimer potentials for the ground states X’C+ and X2X+ of LiH and LiH+, respectively. The potentials used are described in Dalgarno et al.* We include 929 and 64 RV levels for LiH and LiH+, respectively.8 Table 1 gives the maximum N for each v. The molecular constants used are as given in Huber and Herzberg.9 The atomic partition functions were taken from Irwin7 RESULTS
AND DISCUSSION
Tables 2 and 3 list the molecular partition functions and equilibrium constants for LiH and LiH+, respectively. The data can be obtained from the author upon request. The LiH partition functions and dissociation constants are extended beyond the temperature range 1000-9000 K previously given in Ref. 5. The LiH+ data and the LiH ionization constants are presented for the first time. In addition, some results are given for convenient values of 19where 19= 5040/T. The current LiH results differ in comparison to Sauval and Tatem’ for Q and “K by 24 and 42%, respectively at 9000 K. There is generally better agreement for smaller temperatures. As suggested by StanciL6 this discrepancy can be attributed to the approximation
used in Ref. 5. Q,, is the rotational portion of the partition function assuming a separation of electronic, vibrational, and rotational motions and B is the rotational constant. Equation (4) can only be used when B is -2 cm-’ as is found for heavy molecules, but B = 7.5 and 3.8 cm-’ for LiH and LiH+ , respectively. As further suggested by Stanci16 this brings into question the accuracy of all the H-containing molecular constants in Ref. 5. Acknowledgement-This
work was supported
by NSF EPSCoR
grant
OSR-9353227
REFERENCES P. de Bernardis et al, Astron. Astrophy. 269, 1 (1993). R. Maoli, F. Melchiorri, and D. Tosti, Astrophys. J. 425, 372 (1994). M. Signore et al, Astrophys. J. Suppl. 92, 535 (1994). P. C. Stancil, S. Lepp, and A. Dalgarno, Astrophys. J. 458, 401 (1996). A. J. Sauval and J. B. Tatum, Astrophys. J. Suppl. 56, 193 (1984). P. C. Stancil, JQSRT 51, 655 (1994). 7. A. W. Irwin, Astrophys. J. Suppl. 45, 621 (198 1). 8. A. Dalgarno, K. Kirby, and P. C. Stancil, Astrophy. J. 458, 397 (1996). 9. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York (1979).
1. 2. 3. 4. 5. 6.
Radiat. Transfer Vol. 55, No. 6,
pp.849-852,
1996
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
Pergamon !300224073(%)OOO5G8
OO22-4073/96 $15.00 + 0.00
NOTE PARTITION FUNCTIONS CONSTANTS FOR
AND EQUILIBRIUM LiH AND LiH+
P. C. STANCIL W. M. Keck Laboratory for Computational Physics, Department of Physics, University of Nevada Las Vegas, Las Vegas, NV 89154-5002, U.S.A. (Received
24 October
1995)
Abstract-Partition functions and equilibrium constants for the diatomic molecules LiH and LiH+ are presented for temperatures Tup to 40,000 K. The results are obtained by a direct sum of all electronic ground state rotational-vibrational levels whose energies are determined by solving the radial nuclear Schrijdinger equation within the Born-Oppenheimer approximation. Copyright 0 1996 Elsevier Science Ltd
INTRODUCTION Lithium hydride has become a molecule of great interest in studies of the postrecombination era of the early universe. Because of LiH’s large dipole moment, it has been the subject of searches for rotational-vibrational (RV) lines at high redshifts in primordial clouds’ and has been postulated to partially erase spatial anisotropies in the cosmic background radiation (CBR) field by Thomson scattering of CBR photons.* Signore et al3 have suggested that if a large fraction of lithium is contained in LiH, measurements of the LiH RV lines could place constraints on the primordial lithium abundance and discriminate between various big bang nucleosynthesis models. In an analysis of the lithium chemistry of the postrecombination epoch, Stancil et al4 found the abundances of LiH, as well as it’s molecular ion LiH+ , to be so small as to cast doubt on the utility of the above proposals. It may still be possible to observe LiH lines, though, in gravitationally collapsing pre-galactic clouds since the LiH abundance is expected to dramatically increase. During the investigation reported in Ref. 4, discrepancies between the LiH partition functions obtained quantum-mechanically and the polynomial expansions of Sauval and Taturn’ were found. A similar situation for H: and He: was previously pointed out by Stancil.6 New partition functions and dissociation equilibrium constants for LiH and LiH+ and ionization equilibrium constants for LiH are presented.
THEORY
AND
CALCULATIONS
The molecular partition function is given by
& CT) = 1 L’
cN exp[- cEggN
J%N)/h T 1
(1)
for the molecule xH, where x = Li or Li +. The vibrational and rotational quantum numbers are given by u and N, respectively, EcNis the binding energy of the UN RV level, Eg is the binding energy of the lowest RV level, k,, is the Boltzmann constant, and g, is the total statistical weight given by’ 849
Note
850
Table 1. Maximum rotational quantum number N for each vibrational quantum number u of the ground state used in the partition function calculations. Molecule 0
1
2
3
4
V” 5
6
7
8
9
10
11
LiH
57
58
57
56
54
52
50
48
47
45
43
41
LiH+
16
13
11
8
6
4
-
-
-
-
-
-
2,
LiH LiH+
Table
2.
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 252 300 336 400 420 500 504 600 630 700 800 840 900 1000 1008 1100 1200 1260 1300 1400 1500 1600
14
15
16
17
18
19
20
21
22
34
32
30
27
25
22
20
18
15
"KDI
(cm 5
13 36
_
_
_
_
_
_
_
_
Partition functions Q and dissociation “K,, (Dl: LiH-+Li + H) and ionization LiH -+ LiH+ + e-) equilibrium constants for LiH as functions of temperature.
Q 10
12 39 _
1.0431+00 1.3682+00 2.2587+00 3.1881-tOO 4.1254+00 5.0658+00 6.0081-tOO 6.9515+00 7.8957$00 8.8407+00 9.7863+00 1.0732+01 1.1679+01 1.2626+01 1.3574+01 1.4522+01 1.5471+01 1.6420+01 1.7370+01 1.8320+01 1.9271+01 2.4230+01 2.8842+01 3.2339+01 3.8674+01 4.0693+01 4.9010+01 4.9437+01 6.0056+-01 6.3531+01 7.1955+01 8.4803+01 9.0223+01 9.8666+01 1.1359+02 1.1484+02 1.2963+02 1.4680+02 1.5766+02 1.6514+02 1.8468+02 2.0544+02 2.2745+02
-3
Q
"KII
1
1.6175-251 2.7978-204 1.2417-170 1.7613-145 5.4689-126 1.8751-110 8.6112-98 2.7954-87 2.0276-78 7.4714-71 2.5527-64 1.2691-58 1.2952-53 3.5449-49 3.2121-45 1.1401-41 7.1959-28 1.7291-19 8.0772-15 1.1915-08 4.0479-07 2.9829-02 4.7462-02 4.9081+02 4.8824+03 4.7516+05 7.9795+07 4.3748+08 4.2oa9+09 9.9231+10 1.2431+11 1.3065+12 1.1132+13 3.4124+13 6.7937$13 3.1918+14 1.2168+15 3.9153+15
(cm-S)
4.1714-280 9.9453-259 3.3798-240 5.5926-224 1.1528-209 6.1462-197 1.5091-185 2.7067-175 4.6102-135 2.8330-110 2.5635-96 1.0822-77 4.9414-73 4.1033-58 1.7181-57 4.7651-45 6.1670-42 1.0359-35 1.0475-28 2.2661-26 2.9428-23 6.7184-19 1.3758-18 2.4695-15 2.3059-12 8.2838-11 7.5022-10 1.0668-07 7.8031-06 3.3334-04
1680 1700 1800 1900 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000 21000 22000 23000 24000 25000 26000 27000 28000 29000 30000 31000 32000 33000 34000 35000 36000 37000 38000 39000 40000
2.459a+o2 2.5074+02 2.7534+02 3.0127+02 3.2856+02 6.8259+02 1.1986+03 1.8763+03 2.6898+03 3.6013+03 4.5737+03 5.5759+03 6.5844+03 7.5825+03 8.5584+03 9.5049+03 1.0417+04 1.1293$04 1.2131+04 1.2932+04 1.3696+04 1.4425+04 1.5119+04 1.5781+04 1.6411+04 1.7012+04 1.75a5+04 1.8132+04 1.8654+04 1.9153+04 1.9629+04 2.0085+04 2.0522+04 2.0940+04 2.1340+04 2.1724+04 2.2093$04 2.2447+04 2.2788+04 2.3115+04 2.3430+04 2.3733+04 2.4026+04
"KDI
(cm
-3
“K,,
(II:
“Kll
)
9.0086+15 1.0958+16 2.7303+16 6.1691+16 1.2827+17 1.2480+19 1.1571+20 4.3588+20 1.0901+21 2.2274+21 4.1078+21 ‘7.2016+21 1.2352+22 2.1074+22 3.6118-t22 6.2552+22 1.0985+23 1.9603+23 3.5579+23 6.5713+23 1.2350+24 2.3616-t24 4.5923+24 9.0773+24 1.8227+25 3.7161+25 7.6871+25 1.6125+26 3.4278+26 7.3802+26 1.6084+27 3.5463+27 7.9062+27 1.7813+28 4.0541+28 9.3160+28 2.1605+29 5.0543+29 1.1924+30 2.8355+30 6.7942+30 1.6399+31 3.9857+31
(cm-7
4.8641-03 9.1374-03
1.7305-01 2.4006+00 2.5558+01 7.7195+07 1.2527+11 1.0213+13 1.8997+14 1.5349+15 7.4056+15 2.5406+16 6.8726+16 1.5648+17 3.1310+17 5.6712+17 9.4983$17 1.4938+18 2.2318+18 3.1958-tl8 4.4163+18 5.9219+1s 7.7388$-18 9.8912+18 1.2400+19 1.5286+19 1.8564+19 2.2251+19 2.6358+19 3.0896+19 3.5876+19 4.1304+19 4.7188+19 5.3531+19 6.0339+19 6.7614-1-19 7.5359+19 8.3574+19 9.2260+19 1.0142+20 1.1105+20 1.2115+20 1.3171+20
Note Table 3. Partition
functions Q and dissociation equilibrium constants “Ko2 (LiH+ --t Li+ + H) and “Ko3 (LiH+ -+ Li + H+ ) for LiH+ as functions of temperature.
Q 5 10
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 252 300 336 400 420 500 504 600 630 700 BOO 840 900 1000 1008 1100 1200 1260 1300 1400 1500 1600
851
1.3556+00 2.2310+00 4.0722+00 5.9335+00 7.8034+00 9.6804+00 1.1566+01 1.3462+01 1.5378+01 1.7319+01 1.9297+01 2.1321+01 2.3401+01 2.5546+01 2.7767+01 3.0069+01 3.24601-01 3.4945+01 3.7528+01 4.0210+01 4.2995+01 5.9060+01 7.5948+01 8.9572+01 1.1500+02 1.2311+02 1.5557+02 1.5718+02 1.9481+02 2.0611+02 2.3143+02 2.6499+02 2.7755+02 2.9549+02 3.2312+02 3.2521+02 3.4814+02 3.7082+02 3.8342+02 3.9143+02 4.1021+02 4.2737+02 4.4309+02
"KDZ
"KDS
(cmeS)
( cmmS)
Q
5.5155-158
_ _
9.2128-71
3.4684-25 1.6390-09 1.4610-01 9.3099+03 1.5221+07 3.0202+09 1.5929+11 3.4615+12 4.0372+13 2.9919+14 1.5772+15 6.3960+15 2.1106+16 5.9052+16 1.4450+17 3.1669+17 6.3320+17 1.1721+18 2.0323+18 1.6858+19 5.9550+19 1.1938+20 2.9772+20 3.7352+20 7.7185+20 7.9564+20 1.4691+21 1.7173+21 2.3598+21 3.4172+21 3.8820+21 4.6207+21 5.9551+21 6.0672+21 7.4091+21 8.9742+21 9.9636+21 1.0643+22 1.2412+22 1.4274+22 1.6227+22
_ _ _ _ _ _ _ _ 4.2706-284 1.9678-263 2.2652-245 1.8822-229 2.5758-215 1.1132-202 2.5568-191 6.8423-147 1.1684-119 2.2076-104 4.5897-84 5.3954-79 8.5635-63 4.0450-62 1.2056-48 2.7991-45 1.5013-38 5.5706-31 1.8597-28 4.3151-25 2.2363-20 4.8580-20 1.6242-16 2.7038-13 1.3216-11 1.4497-10 3.1910-08 3.4450-06 2.0842-04
1680 1700 1800 1900 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000 21000 22000 23000 24000 25000 26000 27000 28000 29000 30000 31000 32000 33000 34000 35000 36000 37000 38000 39000 40000
4.5475+02 4.5754+02 4.7086+02 4.8316+02 4.9456+02 5.7455+02 6.2014+02 6.4950+02 6.6996+02 6.8503+02 6.9659+02 7.0573+02 7.1314+02 7.1927+02 7.2443+02 7.2882+02 7.3262+02 7.3592+02 7.3882+02 7.4140+02 7.4370+02 7.4576+02 7.4762+02 7.4931+02 7.5085+02 7.5225+02 7.5355+02 7.5474+02 7.5584+02 7.5686+02 7.5781+02 7.5870+02 7.5953+02 7.6030+02 7.6103+02 7.6172+02 7.6236+02 7.6297+02 7.6355+02 7.6409+02 7.6461+02 7.6510+02 7.6556+02
"KDZ
"KD~
( cmw3)
(cm )
1.7852i22 1.8267+22 2.0391+22 2.2597$22 2.4882+22 5.1646+22 8.4482$22 1.2232+23 1.6448+23 2.1051+23 2.6017+23 3.1334+23 3.7000+23 4.3020+23 4.9405+23 5.6172+23 6.3340+23 7.0933+23 7.8977$23 8.7500+23 9.6534+23 1.0611+24 1.1628+24 1.2706+24 1.3851+24 1.5068+24 1.6359+24 1.7732$24 1.9192+24 2.0743+24 2.2393+24 2.4148+24 2.6015+24 2.8001+24 3.0115+24 3.2365+24 3.4759+24 3.7308+24 4.0020+24 4.2908+24 4.5983-i-24 4.9256$24 5.2742+24
-3
3.9206-03 7.8242-03 1.9730-01 3.5578+00 4.8233+01 8.0511+08 3.7576+12 6.6070+14 2.2791+16 3.1431+17 2.4768+18 1.3596+19 5.8519+19 2.1280+20 6.8664+20 2.0328+21 5.6522+21 1.5009+22 3.8532$22 9.6494+22 2.3732+23 5.7619+23 1.3863+24 3.3155124 7.8997+24 1.8787+25 4.4654$25 1.0620+26 2.5292+26 6.0357+26 1.4440+27 3.4649+27 8.3402+27 2.0143128 4.8821+28 1.1875+29 2.8991+29 7.1033+29 1.7468+30 4.3111+30 1.0678+31 2.6539+31 6.6188+31
g, = (2N + 1).
(2)
The dissociation equilibrium constant in terms of number densities can be expressed by “K(T) =
n (x)n (I-0 n (xW -exp(-Di/k,T)
where n(x) is the number density in cme3, QXand QH are the partition functions of the atoms x and H, p is the molecular reduced mass, and 0: is the molecular dissociation energy. A similar expression can be given for the ionization equilibrium constant.
852
Note
To obtain the energies of the RV states, the radial nuclear Schriidinger equation was numerically solved using nonrelativistic Born-Oppenheimer potentials for the ground states X’C+ and X2X+ of LiH and LiH+, respectively. The potentials used are described in Dalgarno et al.* We include 929 and 64 RV levels for LiH and LiH+, respectively.8 Table 1 gives the maximum N for each v. The molecular constants used are as given in Huber and Herzberg.9 The atomic partition functions were taken from Irwin7 RESULTS
AND DISCUSSION
Tables 2 and 3 list the molecular partition functions and equilibrium constants for LiH and LiH+, respectively. The data can be obtained from the author upon request. The LiH partition functions and dissociation constants are extended beyond the temperature range 1000-9000 K previously given in Ref. 5. The LiH+ data and the LiH ionization constants are presented for the first time. In addition, some results are given for convenient values of 19where 19= 5040/T. The current LiH results differ in comparison to Sauval and Tatem’ for Q and “K by 24 and 42%, respectively at 9000 K. There is generally better agreement for smaller temperatures. As suggested by StanciL6 this discrepancy can be attributed to the approximation
used in Ref. 5. Q,, is the rotational portion of the partition function assuming a separation of electronic, vibrational, and rotational motions and B is the rotational constant. Equation (4) can only be used when B is -2 cm-’ as is found for heavy molecules, but B = 7.5 and 3.8 cm-’ for LiH and LiH+ , respectively. As further suggested by Stanci16 this brings into question the accuracy of all the H-containing molecular constants in Ref. 5. Acknowledgement-This
work was supported
by NSF EPSCoR
grant
OSR-9353227
REFERENCES P. de Bernardis et al, Astron. Astrophy. 269, 1 (1993). R. Maoli, F. Melchiorri, and D. Tosti, Astrophys. J. 425, 372 (1994). M. Signore et al, Astrophys. J. Suppl. 92, 535 (1994). P. C. Stancil, S. Lepp, and A. Dalgarno, Astrophys. J. 458, 401 (1996). A. J. Sauval and J. B. Tatum, Astrophys. J. Suppl. 56, 193 (1984). P. C. Stancil, JQSRT 51, 655 (1994). 7. A. W. Irwin, Astrophys. J. Suppl. 45, 621 (198 1). 8. A. Dalgarno, K. Kirby, and P. C. Stancil, Astrophy. J. 458, 397 (1996). 9. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York (1979).
1. 2. 3. 4. 5. 6.