- Email: [email protected]
Ab initio potential energy curve for the J1Δg state of the hydrogen molecule
JOURNAL
OF MOLECULAR
SPECTROSCOPY
91, l28- 136 (1982)
Ab lnitio Potential Energy Curve for the J ‘A, State of the Hydrogen Molecule W. KOEOS Department
of Chemistry,
University
of Warsaw,
Pasteura
I, 02-093
Warsaw,
Poland
AND
J. RYCHLEWSKI Department of Chemistry, The University, Sheffield, S3 7HF, United Kingdom and Institute Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznari. Poland’
of
The Born-Oppenheimer potential energy curve for the J’A, state of the hydrogen molecule was computed using highly flexible wavefunction in the form of a 60-term expansion in elliptic coordinates. The vibrational Schradinger equation for the J state was solved for Hz, HD, and D2. For H2 the resulting energies are compared with the experimental values and it is shown that the adiabatic effects are likely to be responsible for the main part of the existing discrepancy between theoretical and experimental values. For HD the disagreement is too great to be attributable to approximations in the present work and therefore it is suggested that the J state of HD was incorrectly determined.
1.
INTRODUCTION
During the last 20 years the usefulness of the electronic wavefunctions depending explicitly on the interelectronic distance has been established beyond doubt in quantum mechanical calculations for two-electron molecular systems. Using this type of wavefunction and variation method highly accurate Born-Oppenheimer curves for 12 excited states of the Z and II symmetries for hydrogen molecule have been computed (see (I) and references therein). In the present paper the previous approach is extended to the states of A symmetry and the Born-Oppenheimer curve for the J’A, state of H2 is presented. The J state belonging to the 3d singlet XIA complex in the spectrum of HZ has been studied experimentally by Richardson (2) and Dieke (3). However, the experimental data with regard to the vibrational and rotational levels are far from being complete. Dieke and Lewis (4) have measured some of the bands originating from the 3d ‘A state of HD, but these are weak and incomplete. Theoretically the J state has been studied by Browne (5) and by Wakefield and Davidson (6). Browne reported a complete potential curve in the range of R 1.0 < R < 20.0 a.u. Wakefield and Davidson calculated the electronic energy at a ’ Permanent address. 0022-2852/82/010128-09$02.00/O Copyright 0 1982 by Academic
Press. Inc.
All rights of reproduction in any form reserved.
128
POTENTIAL
ENERGY
CURVE
FOR Hz
129
single point near equilibrium distance, which was lower by about 100 cm-’ than Browne’s energy. We shall report below an accurate potential energy curve for the J’A, state calculated in the Born-Oppenheimer approximation using a 60-term wavefunction depending explicitly on the interelectronic distance. Vibrational energies and rotational constants for Hz, HD, and Dz are also calculated and compared with the available experimental values. The nature of the discrepancies is discussed. We hope that results reported here may provide some of the data not available in the literature and stimulate further experimental studies of the J’A, state. 2. POTENTIAL
ENERGY
CURVE
In the present work the James-Coolidge type of function, used previously in the computation of the Born-Oppenheimer potential energy curves for 2 and II states of Hz ( 1), is generalized to represent the electronic wavefunction of the A symmetry. This wavefunction is assumed in the form of the linear combination
where, the + or - sign refers to the singlet or triplet state, respectively, and xl and yl denote the Cartesian coordinates of the Ith electron perpendicular to the molecular axis. The basis functions in elliptic coordinate system, &, vl, have their usual form 4,( 1, 2) = ~-~~l-~~2~;irl~,~~~[[e-8sl-8~2 + (_ I )s,+S;+~@%l+WZ]p~j~
(2)
The value of k in (1) determines the symmetry of the wavefunction; for k = 0, (1) represents a 2: state, k = 1 a II state, and for k = 2 the wavefunction (1) represents a A state. The g or u character of the wavefunction is determined by the parity of Sj + $ + 1 in (2). The remaining symbols have their standard meaning (see, for example, (7)). The selection of terms to the wavefunction was made initially for 30-term expansion for which the four nonlinear parameters (Y,(Y,/I, and j!?have been optimized for five internuclear distances R = 1.0, 1.6, 2.0, 4.0, and 10.0 a.u. For the above values of R, using the optimized exponents, selections of terms have been made, and from the most important terms a 60-term wavefunction has been constructed. At this stage the exponents have been reoptimized. For intermediate internuclear distances the exponents were interpolated. The final Born-Oppenheimer energy curve obtained with a 60-term wavefunction is listed in Table I, where the total energy, E, and the derivative, dE/dR, calculated from the virial theorem, are given in atomic units and the dissociation energy, D, in cm-’ (conversion factor 1 au. of energy = 219 474.62 cm-‘). The J’A, state can be identified as the first member of the 1updS Rydberg series. The diffuse character of 3dli orbital makes the correlation energy for the J state fairly small and this can clear up the small difference of about 20 cm-’ between our value of Born-Oppenheimer energy and that obtained by Wakefield and Davidson for a single point at R = 1.95 a.u. A more complete curve was computed by Browne (5), and is higher than that presented here by about 120 cm-’ in the vicinity of the equilibrium distance. This difference reaches its maximum near R
130
KOLOS AND RYCHLEWSKI TABLE I
Theoretical Energies for the 5’4 State of H2 Computed in the Born-Oppenheimer Using a 60-Term Wavefunction R
E
dE/dR
D
1.0
-0.50720851
-10610.95
-0.52066685
1.3
-0.60805672
11523.89
-0.20139151
1.5
-0.63754538
17995.91
-0.10255895
1.6
-0.64610813
19875.21
-0.07006028
1.7
-0.6518121
21127.09
-0.04506024
1
1.8
-0.65531462
2‘1895.80
-0.02572928
1.9
-0.65710739
22289.27
-0.01072399
1.95
-0.65748654
22372.48
-0.00452956
1.98
-0.65757242
22391.33
-0.00117265
1.99
-0.65757903
22392.78
-0.00010876
2.0
-0.65757529
22391.96
0,00081975
2.05
-0.65740721
22355.07
0.00573636
2.1
-0.65701358
22268.68
0.00996842
2.25
-0.65473310
21768.31
0.01982110
2.5
-0.64843875
20386.72
0.02943365
3.0
-0.63183984
16743.68
0.03491237
3.5
-0.61481507
13007.18
0.03250270
4.0
-0.59977739
9706.79
0.02746932
4.5
-0.58744910
7001.04
0.02188221
5.0
-0.57785699
4895.82
0.01663173
5.5
-0.57070525
3326.19
0.01212118
6.0
-0.56559678
2205.01
0.00849176
6.5
-0.56208299
1433.83
0.00574160
7.0
-0.55974616
920.95
0.00377050
8.0
-0.55126878
377.23
0.00153839
9.0
-0.55627841
159.87
0.00072183
-&555881'J3
72.65
o.ooo23859
10.0
Approximation
= 6.0 a.u. and amounts to about 270 cm-r. For large R our curve is lower by 25 cm-‘. The shape of the energy curve is regular and this is in agreement with Mulliken’s conclusion on the pure lsu 3db character of the J’A, state (8). Obviously it is rather difficult to estimate the accuracy of the present BornOppenheimer energies. The improvement in the dissociation energy obtained by increasing the expansion length from 30 to 60 terms at R = 2.0 a.u. amounts to only 6 cm-‘. A more extensive calculation using more flexible wavefunctions would
POTENTIAL
131
ENERGY CURVE FOR Hz TABLE II
Dissociation Energies for Various Vibrational Levels v
in the J State of Hz (cm-‘)
%
J=U,/\=Q
J=2,A=O
J=2,1\=2
cl
21231.79
21056.33
21173.03
1
19019.36
18853.04
18963.62
2
16933.49
16775.99
16880.71
3
14970.73
14821.79
14920.81
4
13126.83
12986.21
13079.69
5
11399.20
11266.75
11354.79
6
9785.81
9661.43
9744.11
7
8285.66
8169.28
8246.64
8
6897.81
6789.43
6861.45
9
5622.70
5522.38
5589.04
IO
4461.86
4369.80
4430.96
11
3417.77
3334.19
3389.21
12
2494.27
2419.65
2469.17
13
1698.35
1633.35
1676.47
14
1038.91
984.47
1020.53
15
529.04
486.69
514.61
16
185.16
156.84
175.42
certainly lower the energy. On the grounds of our previous experience with BornOppenheimer energies for excited states, however, we believe that present results in the vicinity of the equilibrium distance are accurate within a few cm-‘. 3.
VIBRATIONAL
The vibrational equation
LEVELS
=W-(R) CL 1f(R)
.&I+ l)-112 1 d2 * R2 + E(R) + 2~ dR2
TABLE III Theoretical and Experimental Band Origins for H2 theoretical experimental”
a From Ref. (3) raised by 8.1 cm-’ according to Wilkinson (16).
112426.41 112 467.34 -40.93
(3)
132
KOLOS AND RYCHLEWSKI TABLE IV Vibrational Quanta for the J’A; state of H2 (J = A = 0) (cm-‘) AG(u + ‘/2) Y
theor.
exper.”
A
exper.b
A
0
2212.43 2085.87
2195.71 2071.90
17.12 13.97
2214.7 2088.2
-2.3 -3.7
1
a From Ref. (3), corrected by &A2 using B, from Ref. (13). b From Ref. (13).
with the potential E(R) listed in Table I was solved numerically using Numerov method. The integration step was 0.01 a.u. and the integration out to R = 15.0 a.u. The potential was interpolated using fifth-degree with coefficients determined from the values of E and dE/dR at three
the standard was carried polynomial internuclear
TABLE V Rotational Constants, E,, for the J’A; State of the Hydrogen Molecule (J = A = 0) (cm-‘) v
theor.
0
29.357
29.22
1
27.832
27.51
2
26.358
25.18
3
24.926
4
23.534
5.
22.170
6
20.822
7
19.486
8
18.150
9
16.803
10
15.426
11
14.010
12
12.517
13
10.917
14
9.160
15
7.154
16
4.832
17
2.744
a From reference
13.
exper. a
POTENTIAL
ENERGY CURVE FOR Hz
133
TABLE VI Dissociation Energies (cm-‘)
for VariousVibrationalLevelsin the J’A, State of HD and D2 (J = A = 0)
-7
HD
D2
0
21385.66
21568.72
,
19454.48
19977.06
2
17618.81
18449.60
3
15816.49
16985.07
4
14224.64
15582.15
5
12661.14
14239.21
6
11184.32
12955.23
I
9793.12
11729.15
8
8486.80
10560.45
9
7264.80
9448.61
10
6127.17
8393.32
11
5074.62
7394.16
12
4108.67
6451.24
13 14 15 16
3230.82
5564.81
2444.38
4735.73
1753.93
3964.58
116>.19
3252.42
17
686.18
2601.18
18
327.67
2013.20
19
99.21
1491.45
20
1039.40
distances. Three sets of theoretical dissociation energies for H2 obtained with J = 2 and A = 2, J = 2 and A = 0, J = 0 and A = 0 are given in Table II. The third case is obtained by neglecting the centrifugal term in the potential for nuclear motion in Eq. (3). Experimentally the band origins, & rather than the dissociation energies, can be determined directly. The comparisons between theoretical and experimental band origins are given in Table III. A theoretical value of u8 (J = 0, A = 0) was obtained using the n = 1, n = 3 dissociation limit ( 133 610.00 cm-‘) which was reduced to the Born-Oppenheimer value by adding the difference between the adiabatic correction at R, in the ground state (9) and at R - co in the excited state. The experimental value of band origins was corrected by -&,A2 to correspond to J = 0, A = 0.
134
KOLOS AND RYCHLEWSKI
The difference between the theoretical and experimental values of V: is negative. There are three main factors that can be responsible for this discrepancy: (1) the convergence error in the Born-Oppenheimer computation, (2) adiabatic, and (3) nonadiabatic corrections. The first and third factors give negative contributions to the energy whereas the adiabatic correction to the energy is always positive. The negative value of A& means that the adiabatic correction in the J state is relatively large, larger than that for the ground state. From the A& one can estimate the minimum value of the adiabatic correction for the J state of H2 (for R = R,), which amounts to about 155 cm-‘. Since at R - 00 the adiabatic correction for the J state amounts to about 66 cm- ’ the minimum value of the mass dependent correction to the binding energy should be of the order of -90 cm-‘. The effect can also be larger than the above value since part of it is canceled by the nonadiabatic effect and by further lowering of the Born-Oppenheimer curve. The TABLE VII Rotational Constants, B,, for the J’A, State of HD and D2 (J = A = 0) (cm-‘)
v
HD
0
22.100
14.801
D2
21.105
14.255
.2
20.139
13.725
3
19.197
13.203
4
18.281
12.695
17.383
12.197
16.500
ll.iO6
15.623
11.224
14.756
10.745
6
8 9
13.889
10.271
10
13.018
9.798
11
12.133
9.326
12
11.229
8.853
13
10.300
8.374
14
9.322
7.887
15
8.284
7.394
16
7.161
6.886
17
5.908
6.357
18
4.491
5.802
19
2.888
5.216
20
4.584
POTENTIAL
135
ENERGY CURVE FOR H2 TABLE VIII
Vibrational Quanta for the J’A; State of HD (J = 2, A = 2) (cm-‘) AG(u + %) Y
0
1 2
theor.
exper.”
1929.18 1833.73 1740.44
1832.83
a From Ref. (4).
question, however, arises as to whether such a large value of adiabatic correction is indeed possible for the J state. It is well known that the adiabatic correction contains a so-called “angular term” of the form (10, 11) L(L+
1)-A2 2pR2
’
where L is the quantum number corresponding to the orbital momentum in the united atom and A denotes the quantum number corresponding to the component of the angular momentum along the nuclear axis. If L(L + 1) - A2 # 0 this term diverges like l/R2 at small R and gives a large contribution to the adiabatic correction. The J’A, state can be described as la 3d6 so we have A = 2, L = 2. Hence, term (4) gives a contribution to the adiabatic correction which is twice as large as in the case of C’II, state (A = 1, L = 1) and similar in magnitude to its value in the case of the B’Z: state (L = 1, A = 0). For the latter state the adiabatic correction amounts to 142.4 cm-’ (12) (for R = 2.0 a.u.), which is only slightly smaller than the adiabatic correction for the J state estimated here. In Table IV the theoretical and experimental (corrected to correspond to J = 0, A = 0) values of the vibrational quanta in the J’Ag state of H2 are compared. In this table the vibrational quanta obtained by Ginter (13) are also given. The latter have been corrected for L-uncoupling (extreme A-type doubling) effects. It is seen that the differences between experimental and theoretical vibrational quanta are similar to those for the II, states (14) and they are much smaller when correction for L-uncoupling effect is taken into account. In Table V we compare the theoretical and experimental (corrected in similar way as for vibrational quanta) values of rotational constants. In this case the agreement is quite satisfactory. Vibrational levels and rotational constants for HD and D2 have also been calculated and they are listed in Tables VI and VII. The existing experimental data for HD (4) enable us to make a comparison between experimental and theoretical vt values. The theoretical value of vt obtained in a way similar to that for H2 amounts to 112 548.31 cm-’ whereas the corresponding experimental value obtained using the Dabrowski and Herzberg experimental data for the B’Z: state of HD (15), is 112 519.01 cm-‘. Hence, the disagreement in vt is 29.30 cm-‘, i.e., this difference has the sign opposite to that of the corresponding value for Hz. Also
136
KOLOS AND RYCHLEWSKI
the comparison of the theoretical and experimental vibrational quanta, which is given in Table VIII, leads to an unexpected conclusion. The disagreement in vibrational quanta is of the order of 100 cm-‘, which is much larger than that for Hz and is too great to be attributable to approximations in the present work. It is tentatively suggested therefore that in the very early work on HD the J’A, state was incorrectly determined. The complete lack of experimental data for D2 and the presumably incorrect determination of the J state for HD make impossible a definite estimation of the adiabatic correction for the J state as well as a discussion of the accuracy of the present theoretical results. We hope, however, that the results presented here may stimulate further experimental studies of the J’A, state. ACKNOWLEDGMENT One of the authors (J.R.) is very grateful to Dr. W. T. Raynes for his kind hospitality at Sheffield University, to Dr. G. Herzberg for providing information on experimental results for HD, and to Science Research Council for financial support. Note added in proof: Since this paper was submitted an article by Quadrelli and Dressler [J. Mol. 86, 316-326 (1981)) has appeared in which the large and irregular discrepancies between interpolated and experimental energy levels for the J state of HD have also been found.
Spectrosc.
RECEIVED:
June 29, 198 1 REFERENCES
1. W. Kc~.os, J. Mol. Struct. 46, 73-92 (1978). 2. 0. W. RICHARDSON, “Molecular Hydrogen and Its Spectrum,” Yale Univ. Press, New Haven, 1934. 3. G. H. DIEKE, J. Mol. Spectrosc. 2, 494-517 (1958). 4. G. H. DIEKE AND M. N. LEWIS, Phys. Rev. 52, loo-125 (1937). 5. J. C. BROWNE, J. Chem. Phys. 41, 1583-1586 (1964). 6. C. B. WAKEFIELDAND E. R. DAVIDSON, J. Chem. Phys. 43, 834-839 (1965). 7. W. Kotos AND L. WOLNIEWICZ, J. Chem. Phys. 43, 2429-2441 (1965). 8. R. S. MULLIKEN, J. Amer. Chem. Sot. 88, 1849-1861 (1966). 9. W. Kcnos AND L. WOLNIEWICZ, J. Chem. Phys. 41, 3663-3673 (1964). 10. W. Kcrr.os, Advan. Quantum Chem. 5, 99-133 (1970). 11. A. L. FORD, E. M. GREENAWALT, AND J. C. BROWNE, J. Chem. Phys. 67, 983-988 (1977). 12. W. Kotos AND L. WOLNIEWICZ, J. Chem. Phys. 45, 509-514 (1966). 13. M. L. GINTER, J. Chem. Phys. 46, 3687-3688 (1967). 14. W. Kotos AND J. RYCHLEWSKI,J. Mol. Spectrosc. 66, 428-440 (1977). 15. I. DABROWSKIAND G. HERZBERG,Canad. J. Phys. 54, 525-567 16. P. G. WILKINSON, Canad. J. Phys. 46, 1225-1235 (1968).
(1976).
OF MOLECULAR
SPECTROSCOPY
91, l28- 136 (1982)
Ab lnitio Potential Energy Curve for the J ‘A, State of the Hydrogen Molecule W. KOEOS Department
of Chemistry,
University
of Warsaw,
Pasteura
I, 02-093
Warsaw,
Poland
AND
J. RYCHLEWSKI Department of Chemistry, The University, Sheffield, S3 7HF, United Kingdom and Institute Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznari. Poland’
of
The Born-Oppenheimer potential energy curve for the J’A, state of the hydrogen molecule was computed using highly flexible wavefunction in the form of a 60-term expansion in elliptic coordinates. The vibrational Schradinger equation for the J state was solved for Hz, HD, and D2. For H2 the resulting energies are compared with the experimental values and it is shown that the adiabatic effects are likely to be responsible for the main part of the existing discrepancy between theoretical and experimental values. For HD the disagreement is too great to be attributable to approximations in the present work and therefore it is suggested that the J state of HD was incorrectly determined.
1.
INTRODUCTION
During the last 20 years the usefulness of the electronic wavefunctions depending explicitly on the interelectronic distance has been established beyond doubt in quantum mechanical calculations for two-electron molecular systems. Using this type of wavefunction and variation method highly accurate Born-Oppenheimer curves for 12 excited states of the Z and II symmetries for hydrogen molecule have been computed (see (I) and references therein). In the present paper the previous approach is extended to the states of A symmetry and the Born-Oppenheimer curve for the J’A, state of H2 is presented. The J state belonging to the 3d singlet XIA complex in the spectrum of HZ has been studied experimentally by Richardson (2) and Dieke (3). However, the experimental data with regard to the vibrational and rotational levels are far from being complete. Dieke and Lewis (4) have measured some of the bands originating from the 3d ‘A state of HD, but these are weak and incomplete. Theoretically the J state has been studied by Browne (5) and by Wakefield and Davidson (6). Browne reported a complete potential curve in the range of R 1.0 < R < 20.0 a.u. Wakefield and Davidson calculated the electronic energy at a ’ Permanent address. 0022-2852/82/010128-09$02.00/O Copyright 0 1982 by Academic
Press. Inc.
All rights of reproduction in any form reserved.
128
POTENTIAL
ENERGY
CURVE
FOR Hz
129
single point near equilibrium distance, which was lower by about 100 cm-’ than Browne’s energy. We shall report below an accurate potential energy curve for the J’A, state calculated in the Born-Oppenheimer approximation using a 60-term wavefunction depending explicitly on the interelectronic distance. Vibrational energies and rotational constants for Hz, HD, and Dz are also calculated and compared with the available experimental values. The nature of the discrepancies is discussed. We hope that results reported here may provide some of the data not available in the literature and stimulate further experimental studies of the J’A, state. 2. POTENTIAL
ENERGY
CURVE
In the present work the James-Coolidge type of function, used previously in the computation of the Born-Oppenheimer potential energy curves for 2 and II states of Hz ( 1), is generalized to represent the electronic wavefunction of the A symmetry. This wavefunction is assumed in the form of the linear combination
where, the + or - sign refers to the singlet or triplet state, respectively, and xl and yl denote the Cartesian coordinates of the Ith electron perpendicular to the molecular axis. The basis functions in elliptic coordinate system, &, vl, have their usual form 4,( 1, 2) = ~-~~l-~~2~;irl~,~~~[[e-8sl-8~2 + (_ I )s,+S;+~@%l+WZ]p~j~
(2)
The value of k in (1) determines the symmetry of the wavefunction; for k = 0, (1) represents a 2: state, k = 1 a II state, and for k = 2 the wavefunction (1) represents a A state. The g or u character of the wavefunction is determined by the parity of Sj + $ + 1 in (2). The remaining symbols have their standard meaning (see, for example, (7)). The selection of terms to the wavefunction was made initially for 30-term expansion for which the four nonlinear parameters (Y,(Y,/I, and j!?have been optimized for five internuclear distances R = 1.0, 1.6, 2.0, 4.0, and 10.0 a.u. For the above values of R, using the optimized exponents, selections of terms have been made, and from the most important terms a 60-term wavefunction has been constructed. At this stage the exponents have been reoptimized. For intermediate internuclear distances the exponents were interpolated. The final Born-Oppenheimer energy curve obtained with a 60-term wavefunction is listed in Table I, where the total energy, E, and the derivative, dE/dR, calculated from the virial theorem, are given in atomic units and the dissociation energy, D, in cm-’ (conversion factor 1 au. of energy = 219 474.62 cm-‘). The J’A, state can be identified as the first member of the 1updS Rydberg series. The diffuse character of 3dli orbital makes the correlation energy for the J state fairly small and this can clear up the small difference of about 20 cm-’ between our value of Born-Oppenheimer energy and that obtained by Wakefield and Davidson for a single point at R = 1.95 a.u. A more complete curve was computed by Browne (5), and is higher than that presented here by about 120 cm-’ in the vicinity of the equilibrium distance. This difference reaches its maximum near R
130
KOLOS AND RYCHLEWSKI TABLE I
Theoretical Energies for the 5’4 State of H2 Computed in the Born-Oppenheimer Using a 60-Term Wavefunction R
E
dE/dR
D
1.0
-0.50720851
-10610.95
-0.52066685
1.3
-0.60805672
11523.89
-0.20139151
1.5
-0.63754538
17995.91
-0.10255895
1.6
-0.64610813
19875.21
-0.07006028
1.7
-0.6518121
21127.09
-0.04506024
1
1.8
-0.65531462
2‘1895.80
-0.02572928
1.9
-0.65710739
22289.27
-0.01072399
1.95
-0.65748654
22372.48
-0.00452956
1.98
-0.65757242
22391.33
-0.00117265
1.99
-0.65757903
22392.78
-0.00010876
2.0
-0.65757529
22391.96
0,00081975
2.05
-0.65740721
22355.07
0.00573636
2.1
-0.65701358
22268.68
0.00996842
2.25
-0.65473310
21768.31
0.01982110
2.5
-0.64843875
20386.72
0.02943365
3.0
-0.63183984
16743.68
0.03491237
3.5
-0.61481507
13007.18
0.03250270
4.0
-0.59977739
9706.79
0.02746932
4.5
-0.58744910
7001.04
0.02188221
5.0
-0.57785699
4895.82
0.01663173
5.5
-0.57070525
3326.19
0.01212118
6.0
-0.56559678
2205.01
0.00849176
6.5
-0.56208299
1433.83
0.00574160
7.0
-0.55974616
920.95
0.00377050
8.0
-0.55126878
377.23
0.00153839
9.0
-0.55627841
159.87
0.00072183
-&555881'J3
72.65
o.ooo23859
10.0
Approximation
= 6.0 a.u. and amounts to about 270 cm-r. For large R our curve is lower by 25 cm-‘. The shape of the energy curve is regular and this is in agreement with Mulliken’s conclusion on the pure lsu 3db character of the J’A, state (8). Obviously it is rather difficult to estimate the accuracy of the present BornOppenheimer energies. The improvement in the dissociation energy obtained by increasing the expansion length from 30 to 60 terms at R = 2.0 a.u. amounts to only 6 cm-‘. A more extensive calculation using more flexible wavefunctions would
POTENTIAL
131
ENERGY CURVE FOR Hz TABLE II
Dissociation Energies for Various Vibrational Levels v
in the J State of Hz (cm-‘)
%
J=U,/\=Q
J=2,A=O
J=2,1\=2
cl
21231.79
21056.33
21173.03
1
19019.36
18853.04
18963.62
2
16933.49
16775.99
16880.71
3
14970.73
14821.79
14920.81
4
13126.83
12986.21
13079.69
5
11399.20
11266.75
11354.79
6
9785.81
9661.43
9744.11
7
8285.66
8169.28
8246.64
8
6897.81
6789.43
6861.45
9
5622.70
5522.38
5589.04
IO
4461.86
4369.80
4430.96
11
3417.77
3334.19
3389.21
12
2494.27
2419.65
2469.17
13
1698.35
1633.35
1676.47
14
1038.91
984.47
1020.53
15
529.04
486.69
514.61
16
185.16
156.84
175.42
certainly lower the energy. On the grounds of our previous experience with BornOppenheimer energies for excited states, however, we believe that present results in the vicinity of the equilibrium distance are accurate within a few cm-‘. 3.
VIBRATIONAL
The vibrational equation
LEVELS
=W-(R) CL 1f(R)
.&I+ l)-112 1 d2 * R2 + E(R) + 2~ dR2
TABLE III Theoretical and Experimental Band Origins for H2 theoretical experimental”
a From Ref. (3) raised by 8.1 cm-’ according to Wilkinson (16).
112426.41 112 467.34 -40.93
(3)
132
KOLOS AND RYCHLEWSKI TABLE IV Vibrational Quanta for the J’A; state of H2 (J = A = 0) (cm-‘) AG(u + ‘/2) Y
theor.
exper.”
A
exper.b
A
0
2212.43 2085.87
2195.71 2071.90
17.12 13.97
2214.7 2088.2
-2.3 -3.7
1
a From Ref. (3), corrected by &A2 using B, from Ref. (13). b From Ref. (13).
with the potential E(R) listed in Table I was solved numerically using Numerov method. The integration step was 0.01 a.u. and the integration out to R = 15.0 a.u. The potential was interpolated using fifth-degree with coefficients determined from the values of E and dE/dR at three
the standard was carried polynomial internuclear
TABLE V Rotational Constants, E,, for the J’A; State of the Hydrogen Molecule (J = A = 0) (cm-‘) v
theor.
0
29.357
29.22
1
27.832
27.51
2
26.358
25.18
3
24.926
4
23.534
5.
22.170
6
20.822
7
19.486
8
18.150
9
16.803
10
15.426
11
14.010
12
12.517
13
10.917
14
9.160
15
7.154
16
4.832
17
2.744
a From reference
13.
exper. a
POTENTIAL
ENERGY CURVE FOR Hz
133
TABLE VI Dissociation Energies (cm-‘)
for VariousVibrationalLevelsin the J’A, State of HD and D2 (J = A = 0)
-7
HD
D2
0
21385.66
21568.72
,
19454.48
19977.06
2
17618.81
18449.60
3
15816.49
16985.07
4
14224.64
15582.15
5
12661.14
14239.21
6
11184.32
12955.23
I
9793.12
11729.15
8
8486.80
10560.45
9
7264.80
9448.61
10
6127.17
8393.32
11
5074.62
7394.16
12
4108.67
6451.24
13 14 15 16
3230.82
5564.81
2444.38
4735.73
1753.93
3964.58
116>.19
3252.42
17
686.18
2601.18
18
327.67
2013.20
19
99.21
1491.45
20
1039.40
distances. Three sets of theoretical dissociation energies for H2 obtained with J = 2 and A = 2, J = 2 and A = 0, J = 0 and A = 0 are given in Table II. The third case is obtained by neglecting the centrifugal term in the potential for nuclear motion in Eq. (3). Experimentally the band origins, & rather than the dissociation energies, can be determined directly. The comparisons between theoretical and experimental band origins are given in Table III. A theoretical value of u8 (J = 0, A = 0) was obtained using the n = 1, n = 3 dissociation limit ( 133 610.00 cm-‘) which was reduced to the Born-Oppenheimer value by adding the difference between the adiabatic correction at R, in the ground state (9) and at R - co in the excited state. The experimental value of band origins was corrected by -&,A2 to correspond to J = 0, A = 0.
134
KOLOS AND RYCHLEWSKI
The difference between the theoretical and experimental values of V: is negative. There are three main factors that can be responsible for this discrepancy: (1) the convergence error in the Born-Oppenheimer computation, (2) adiabatic, and (3) nonadiabatic corrections. The first and third factors give negative contributions to the energy whereas the adiabatic correction to the energy is always positive. The negative value of A& means that the adiabatic correction in the J state is relatively large, larger than that for the ground state. From the A& one can estimate the minimum value of the adiabatic correction for the J state of H2 (for R = R,), which amounts to about 155 cm-‘. Since at R - 00 the adiabatic correction for the J state amounts to about 66 cm- ’ the minimum value of the mass dependent correction to the binding energy should be of the order of -90 cm-‘. The effect can also be larger than the above value since part of it is canceled by the nonadiabatic effect and by further lowering of the Born-Oppenheimer curve. The TABLE VII Rotational Constants, B,, for the J’A, State of HD and D2 (J = A = 0) (cm-‘)
v
HD
0
22.100
14.801
D2
21.105
14.255
.2
20.139
13.725
3
19.197
13.203
4
18.281
12.695
17.383
12.197
16.500
ll.iO6
15.623
11.224
14.756
10.745
6
8 9
13.889
10.271
10
13.018
9.798
11
12.133
9.326
12
11.229
8.853
13
10.300
8.374
14
9.322
7.887
15
8.284
7.394
16
7.161
6.886
17
5.908
6.357
18
4.491
5.802
19
2.888
5.216
20
4.584
POTENTIAL
135
ENERGY CURVE FOR H2 TABLE VIII
Vibrational Quanta for the J’A; State of HD (J = 2, A = 2) (cm-‘) AG(u + %) Y
0
1 2
theor.
exper.”
1929.18 1833.73 1740.44
1832.83
a From Ref. (4).
question, however, arises as to whether such a large value of adiabatic correction is indeed possible for the J state. It is well known that the adiabatic correction contains a so-called “angular term” of the form (10, 11) L(L+
1)-A2 2pR2
’
where L is the quantum number corresponding to the orbital momentum in the united atom and A denotes the quantum number corresponding to the component of the angular momentum along the nuclear axis. If L(L + 1) - A2 # 0 this term diverges like l/R2 at small R and gives a large contribution to the adiabatic correction. The J’A, state can be described as la 3d6 so we have A = 2, L = 2. Hence, term (4) gives a contribution to the adiabatic correction which is twice as large as in the case of C’II, state (A = 1, L = 1) and similar in magnitude to its value in the case of the B’Z: state (L = 1, A = 0). For the latter state the adiabatic correction amounts to 142.4 cm-’ (12) (for R = 2.0 a.u.), which is only slightly smaller than the adiabatic correction for the J state estimated here. In Table IV the theoretical and experimental (corrected to correspond to J = 0, A = 0) values of the vibrational quanta in the J’Ag state of H2 are compared. In this table the vibrational quanta obtained by Ginter (13) are also given. The latter have been corrected for L-uncoupling (extreme A-type doubling) effects. It is seen that the differences between experimental and theoretical vibrational quanta are similar to those for the II, states (14) and they are much smaller when correction for L-uncoupling effect is taken into account. In Table V we compare the theoretical and experimental (corrected in similar way as for vibrational quanta) values of rotational constants. In this case the agreement is quite satisfactory. Vibrational levels and rotational constants for HD and D2 have also been calculated and they are listed in Tables VI and VII. The existing experimental data for HD (4) enable us to make a comparison between experimental and theoretical vt values. The theoretical value of vt obtained in a way similar to that for H2 amounts to 112 548.31 cm-’ whereas the corresponding experimental value obtained using the Dabrowski and Herzberg experimental data for the B’Z: state of HD (15), is 112 519.01 cm-‘. Hence, the disagreement in vt is 29.30 cm-‘, i.e., this difference has the sign opposite to that of the corresponding value for Hz. Also
136
KOLOS AND RYCHLEWSKI
the comparison of the theoretical and experimental vibrational quanta, which is given in Table VIII, leads to an unexpected conclusion. The disagreement in vibrational quanta is of the order of 100 cm-‘, which is much larger than that for Hz and is too great to be attributable to approximations in the present work. It is tentatively suggested therefore that in the very early work on HD the J’A, state was incorrectly determined. The complete lack of experimental data for D2 and the presumably incorrect determination of the J state for HD make impossible a definite estimation of the adiabatic correction for the J state as well as a discussion of the accuracy of the present theoretical results. We hope, however, that the results presented here may stimulate further experimental studies of the J’A, state. ACKNOWLEDGMENT One of the authors (J.R.) is very grateful to Dr. W. T. Raynes for his kind hospitality at Sheffield University, to Dr. G. Herzberg for providing information on experimental results for HD, and to Science Research Council for financial support. Note added in proof: Since this paper was submitted an article by Quadrelli and Dressler [J. Mol. 86, 316-326 (1981)) has appeared in which the large and irregular discrepancies between interpolated and experimental energy levels for the J state of HD have also been found.
Spectrosc.
RECEIVED:
June 29, 198 1 REFERENCES
1. W. Kc~.os, J. Mol. Struct. 46, 73-92 (1978). 2. 0. W. RICHARDSON, “Molecular Hydrogen and Its Spectrum,” Yale Univ. Press, New Haven, 1934. 3. G. H. DIEKE, J. Mol. Spectrosc. 2, 494-517 (1958). 4. G. H. DIEKE AND M. N. LEWIS, Phys. Rev. 52, loo-125 (1937). 5. J. C. BROWNE, J. Chem. Phys. 41, 1583-1586 (1964). 6. C. B. WAKEFIELDAND E. R. DAVIDSON, J. Chem. Phys. 43, 834-839 (1965). 7. W. Kotos AND L. WOLNIEWICZ, J. Chem. Phys. 43, 2429-2441 (1965). 8. R. S. MULLIKEN, J. Amer. Chem. Sot. 88, 1849-1861 (1966). 9. W. Kcnos AND L. WOLNIEWICZ, J. Chem. Phys. 41, 3663-3673 (1964). 10. W. Kcrr.os, Advan. Quantum Chem. 5, 99-133 (1970). 11. A. L. FORD, E. M. GREENAWALT, AND J. C. BROWNE, J. Chem. Phys. 67, 983-988 (1977). 12. W. Kotos AND L. WOLNIEWICZ, J. Chem. Phys. 45, 509-514 (1966). 13. M. L. GINTER, J. Chem. Phys. 46, 3687-3688 (1967). 14. W. Kotos AND J. RYCHLEWSKI,J. Mol. Spectrosc. 66, 428-440 (1977). 15. I. DABROWSKIAND G. HERZBERG,Canad. J. Phys. 54, 525-567 16. P. G. WILKINSON, Canad. J. Phys. 46, 1225-1235 (1968).
(1976).