J. Phys. Chem. 1982, 86, 3592-3598
3592
Electronic Structure of Rydberg States of H3, NeH, H,F, H,O, NH4, and CH, Molecules Susanne Raynor and Dudley R. Herschbach’ wmt of -try.
tfarvard Universny, C%mbr&e, I n Final Form: May 28, 1982)
Messeahusetts 02138 ( R e c e M : February 25, 1982;
Six exemplary m o l d e a are treated for which the ground electronic state is diasocitive whereas the corresponding cation is quite stable. Such molecules posaess long-Qvedexcited Rydberg states with a quasi-hydrogenicelectron orbiting outside the cationic core. Electronic energies and spectroscopic transition moments are calculated by ab initio methods employing a floating spherical Slater orbital (FSSO) variational function. For H3, the results agree satisfactorily with other calculations. For the five second-row hydride molecules, strong transitions are predicted to occur in two regions: 5OOCF7000A and 9000-12 0oO A. Comparison of our results for H3 and NH4 with the experimental spectra of Herzberg shows excellent agreement for some transitions, but large discrepancies in wavelength and especially in intensities appear for other transitions.
Introduction The discovery of Rydberg spectra for H3 by Herzberg1I2 calls attention to a large class of molecules for which the ground electronic state is dissociative but excited states enjoy lifetimes quite long compared to vibrational and rotational periods. For such molecules, the corresponding cation is stable. Rydberg states with sufficient excitation can then be described approximately as quasi-hydrogenic, with the spherical atomic core replaced by a molecular cation which imposes “internal crystal field” splittings appropriate to its symmetry. Ab initio electronic structure for H3 which employ a simple “frozen core approximation” have indeed given good agreement with most features of the observed spectra. However, sone discrepancies between theory and experiment still persist. This paper applies similar methods to the five secondrow hydrides, NeH, H2F, H30, NH4, and CH6. Our chief aim is to provide results that may aid the assignment and analysis of spectra for these molecules. Of particular interest is NH4, for which Herzberg has recently identified two band systems! Another motivation came from Figure 1, which compares the Rydberg energy levels of H3 with those for Li, corresponding to the united atom limit. A curious pattern of “symmetry exchange” appears: the (np)E’ and (ns)A,’ states of H3 are remarkably close in energy to the ns and np states of Li, respectively. Comparison of Rydberg states of H2 and He reveals an analogous pattern.’ This prompted us to look for similar patterns in the Rydberg states of the second-row hydrides. We have not found such “symmetry exchange” there but do observe trends in quantum defects and splitting parameters that should prove useful in searching for and interpreting spectra of this family of metastable molecules. Computational Methods We adopted the strategy described for H3 by King and M ~ r o k u m a .A~ basis set of core orbitals was selected to represent the parent ion, and the geometry and orbital exponents were optimized. Rydberg-like orbitals centered on the second-row atom were than added. The virtual (1) Herzberg, G. J. Chem. Phys. 1979, 70, 4806. (2) Dabrowski, 1.; Henberg, G. Can. J.Phys. 1980,68,1238. Henberg, G.; Watson, J. K. G. Can. J. Phys. 1980,58,1250. Henberg, G.; Lew, H.; Sloan, J. J.; Watson, J. K. G. Can. J. Phys. 1981,59, 428. (3) King, H. F.; Morokuma, K. J. Chem. Phys. 1979, 71, 3213. (4) Jungen, M. J. Chem. Phys. 1979, 71, 3540. (5) Martin, R. L.J. Chem. Phys. 1979, 71, 3541. (6) Henberg, G. Discuss. Faraday Soc. 1981, 71, 165. (7) Huber, K. P.; Herzberg, G. ‘Molecule Spectra and Molecular Structure, Constants of Diatomic Molecules”;van Nonatrand-Reinhold New York, 1979. 0022-3654/82/2086-3592$01.25/0
orbital energies ek were determined and added to the energy E, of the parent ion to obtain the Rydberg energies Ek of the molecule via Koopmans’ theorem. Choice of &bit&. Slater-type orbitals (STO)were used for all of the atomic orbitals rather than the Gaussian functions employed in the previous H3calculations.3a This permitted use of the highly efficient POLCAL program of Stevens.8 STOs offer the advantage that fewer orbitals we required for convergence to a good representation of the parent ion. The long-range behavior is also better than for Gauasians, and this becomes significantwhen modeling the higher Rydberg orbitals. The chief disadvantage of STOs is the increased difficulty of computing multicenter integrals, particularly when two or more orbitals with nonzero angular momentum appear on different centers. Also, evaluation of these integrals becomes more time consuming as the STO increases in size, a decided shortcoming in this application. Accordingly, we chose to limit the use of p and d orbitals to the second-row atom. For the core orbitals of the ion we tried two kinds of basis sets. The first was a minimum basis set (MBS) with Clementi-Raimondi exponentsQon the heavy atom and optimized 1s orbitals on the hydrogens. The second we refer to as 4 floating spherical Slater orbital (FSSO)basis set. It contains (in addition to the MBS) floating 1s orbitals located between each hydrogen pair, with exponents optimized along with those for the orbitals OIJ the hydrogens. As with analogous floating Gaussian orbital schemes,lo this provides for polarization between hydrogen atoms without installing p or d orbitals on the hydrogens. We find the FSSO basis sets dQindeed lower the SCF energy of the parent ions significantly (cf. Table I). For H3, King and Morokuma3 estimated that the limiting Hartree-Fock energy is -1.300 36 hartrees. By the simple expedient of adding the floating 1s orbitals, we obtain a res& (-1.295443)corresponding to 86% of the difference between the minimum basis set and the Hartree-Fock limit and thereby avoid the complexity introduced by off-center orbitals with nonzero angular momentum. Procedure. For simplicity, we chose to optimize the bond lengths and angles of the parent ion and initial 1s exponents on the hydrogens using a MBS calculations. For the geometry thus determined (cf. Table I and Figure 2), we employed the FSSO basis to carry out a simultaneous optimization of the 1s exponents on both the hydrogens (8) Porter, R. N. Stavens, R. M.; Karplus, M. J.Chem. Phys. 1968,49, 5163. (9) Clementi, E.; Raimondi, D. L. J. Chem. Phys. 1963, 38, 2686. (10) Frost, A. A. J. Chem. Phys. 1967,47, 3707.
0 1982 American Chemlcal Society
The Journal of Physical Chemistry. Vol. 86, No. 18, 1982 3593 Li 0.00
s
-
H, 4
Li
H,
Li
P
A;
D
I
H,
E'
I
I
I
9
A;
I
I
3s
3p 3d
.
.. ..
I
I
I
4s
I
I
4P 4d
E" E' n
1
H3O
,
.
.
..
.
. .
I .I I
&-o
.. ..
I' 'I Figuo 1. Cwelatlon of energy levels of Ll atom wlth Rydberg states of H&,,). Energies are glvm relathre to the khired a m or molecule. In each case,symmetry spedes of the molecular levels are indicated.
.
.
I
I
TABLE I: Optimum Geometries, Exponents, and Energies for Ions molecule/ eeometr?
basis setb
MBS H3' ( D s h ) ( R H H= 1.6401) FSSO FSSO MBS NeH' (C, (RN~H = 1 . 7 9 4 ) FSSO HP' (C2U) ( R F H= 1.790, LHFH = 115.3) (C3U
1
( R O H = 1.824,
FSSO
+
-1.296 069 -128.002 576 -128.088 571
e
1.413 1.338, 1.018
-76.070643 -76.083 322
e
-76.139 176
1.392 1.286, 1.063 Ryd e
- 56.418 361 -56.441 928
SMBS SMBS + Ryd e
-99.917 587
- 56.458 864
-39.389 340 -39.786 782
Bond len hs in bohrs, bond angles in degrees, energy MBS, minimum basis set; F&O, floatingin hartrees. spherical-Slater-orbital basis set; FSSO + Ryd, FSSO + Rydberg-like orbitals on heavy atom; SMBS, minimum Hydrogen basis set for C atom, no orbitals on H atoms. exponent followed by "floating" orbital exponent. Rydberg basis = 5 each of 2s, 2p, and 3d orbitals with exponents of 1.6, 0.8, 0.4, 0.2, 0.1. e Rydberg basis = 5 each of 3s, 3p, and 3d orbitals with exponents of 1.6, 0.8, 0.4, 0.2, 0.1. f Geometry from ref 19a: R C H= 2.04 a,,, LHCH = 111" for CH, group and R C H= 2.31 a,, LHCH = 47" for CH, group.
and the floating orbitals. Finally, with the geometry and these 1s exponents fixed, we added s, p, and d Slater orbitals to the second row atom to construct the molecular Rydberg orbitals. Five each of 3s,3p, and 3d orbitals were employed, with exponents of 1.6, 0.8, 0.4, 0.2, and 0.1, spanning four octaves. For all our calculations, this basis set yielded energies consistent to within 2 x lW3 hartrees! when compared with results from a three-octave basis lacking the most diffuse orbitals with 0.1 exponents. Energy levels Ekwere computed for all Rydberg states with united atom limits up to 5s,4p, and 4d. The corresponding quantum defecta dkwere determined from E k = -'Iz(n - dk)-2, where n is the principal united atom N
I 0
-99.828 536 -99.834 758
+ Ryd
MBS FSSO FSSO
CH,' ( C * 4
+
1.481
enerm" -1.270 120 -1.295 443
1.426 1.420, 0.996 Ryd e
MBS FSSO
LHOH = 116.0) NH,' ( T d ) (RNH = 1.929)
1.395 1.458, 1.311 Ryd d
+ Ryd
MBS FSSO FSSO
H3O'
+
exponentsC
I
I
I
I
I
10
I
I
I 20
< r > (bohr) Flgure 2. Average radii, ( r ) , for the lowest Rydberg states of the flrstarKl SBCond-Iow hydrides. Levels with the same united atom limit are connected by dotted lines wlth the united atom limit indicated at the base. The molecular geometries are shown schematically with the heavy atoms denoted by a fllled ckcle and the hydrogen atoms by open circles.
quantum number for the level. Since some important emission bands may arise from still higher Rydberg states, energy levels for such states were estimated by assuming the quantum defect to be equal to dk for the highest computed state with the same symmetry and the same orbital angular momentum in the united &tomlimit. In order to predict the intensity distribution in the emission spectrum, we also estimated the Einstein coefficients A, for spontaneous emissionll for each transition considered (with upper state i and lower state f). The required transition dipole momenta were evaluated directly from the ab initio wave functions. For the upper state, we used two different forms for I)i depending on the united atom limit of the upper state. If the imited atom limit was 55, 4p, 4d, or a lower-lying level, we used the molecular orbital from the FSSO calculation. If the united atom limit corresponded to higher excitation, we used a hydrogenic approximation for the upper Rydberg state, with the orbial exponent given by uk = l / ( n - dk). For the lower state, we used for I)f the one-electron molecular orbital determined from the FSSO calculation. Error Estimates. The energy levels are subject to four principal kinds of error. (1)The limited size of the basis set used from the parent ion should matter most for the lowest-lying bound state of the molecule and the lowest s-type Rydberg orbital. There the orbital size and extent is smallest and interactions with the hydrogen atoms are largest. (2) The limited size of the Rydberg basis for the second-row atom should matter most for the highest Rydberg molecular orbital states. (3) Configuration interaction is neglected, but the correlation energies for all (11) Herzbprg, G. "Molecular Spectra and Molecular Structure 111. Electronic Spectra and Electronic Structure of Polyatomic Molecules"; Van Nostrand New York, 1966.
3594
The Journal of phvsical Chemistry, Vol. 86,No. 18, 1982
TABLE 11: Quantum Defects of Li and H, Rydberg Levelsa
Li (ref 13) 2s 3s
S 0.4145 0.4038
2p
0.0406
3p
0.0444
3d
H, (ref 3)
A,'
0.0836 0.0706
H3
DIM
(this work)
(ref 12)
A,'
A,'
0.0731 0.0390
0.117 0.160
A,", E' 0.0165 0.4068 0.0229 0.3490
A,", E'
D
A , ' , E", E'
A,', E", E'
A , ' , E", E'
0.0014
-0.0151 -0.0064 0.0216
-0.0384 -0.0294 - 0.0002
0.091 -0.162 -0.162
P
A,", E' 0.096 0.438 0.089 0.950
0.01 22 0.4072 0.0030 0.3283
a The energy levels are given in hartree units by E k =
-I/,(n - dk)-', with d k the tabulated quantum defect. Zero of energy corresponds to electron plus parent ion at infinite separation. Symmetries of levels in order of increasing stability are indicated at top of each column.
accessible Rydberg states are expected3 to be nearly equivalent. (4) The geometry of the parent ion found by using the minimum basis set will not be optimum for the Rydberg states. This discrepancy may become substantial for the s-type Rydberg states, since for these the vertex atom orbitals are likely to become most strongly mixed with the hydrogen atom orbitals. As a test, we used the FSSO procedure to compute energy levels for the lowest Rydberg states of H3 (up to 38, 3p, and 3d united atom states) for comparison with results obtained by King and Morokuma with a large Gaussian basis seta3 Table I1 gives the comparison in terms of quantum defects. We find agreement within 1 X 10-3-1.5 X hartrees; the largest disparity occurs for the (2s)A1' state. As expected, the results me much better than found for the semiempirical diatomics-in-molecules method.12 We also computed energies for the lowest Rydberg states of Na (up to 5s,4p, and 4d) for comparison with experimental data.13 The errors are larger for Na, primarily because the lack of hydrogen atoms eliminates the floating orbitals from the basis. The largest error is 6 X hartrees and occurs for the 3s level. The next largest errors are -3 X lo9 hatrees, for p-type orbitals. Thus for both H3 and Na the largest errors occur for the lowest s-type states, as expected. One previous calculation for the Rydberg states of NH4 has been published by Strehl et They did a minimum basis set restricted Hartree-Fock calculation without including atomic orbitals on the hydrogen atoms. The agreement with our calculated orbital energies is remarkable. The largest difference occurs for the ground (%)AI state which we predict to be more stable by 8 X hartrees. The disagreements for the higher Rydberg states are considerably reduced, with differences of 3 X hartrees or less. After completing our calculations, we learned of similar work on NH4 undertaken by Havriliak and King.15 For all levels with p and d united atoms, our orbital energies agree with theirs to within 1 x hartrees or better. For levels with s united atoms, the discrepancies are larger, as expected. These are 2 X 3X and 4 X for
-
(12)Raynor, S.;Herschbach, D. R. J . Phys. Chem. 1982,86, 1214. (13)Moore C. E.Natl. Stand. Ref. Data Sec., Natl. Bur. Stand. 1971, no. 35. (14)Strehl, W.;Hartmann, H.; Henson, K Sarholz, W.Theor. Chim. Acta 1970,18, 290. (15)Havriliak, S.; King, H. F., private communication.
Raynor and Herschbach
the (3s)-, (4s)-, and (5s)A1' levels, respectively. The effect of deviations from the nominal geometry was examined by recalculating the energy levels for NH4with variations of f0.05 bohr in the NH bond length. The resultant energies were then fit to a quadratic in r, from which the optimal bond length and energy for each level were evaluated. These were compared with the nominal case (NH bond length of 1.929 bohr). The optimum bond lengths for each level were found to differ by no more than d=O.O1 bohr and the optimum energies by no more than 0.2 X hartrees. This indicates that use of the nominal geometry obtained with the minimum basis set does not introduce significant errors. Thus we expect that errors in energies calculated with our FSSO + Rydberg basis set will lie within f 3 X to f 5 X hartrees, with the largest errors occurring for the lowest, unbound s-type states. For the bound Rydberg states of chief interest here, typical errors are expected to be -3 X hartrees or less.
Results and Discussion Table I gives parameters characterizing the parent ions, as computed by use of the indicated basis sets. Figure 2 shows to scale the geometries of the ion cores and orbits corresponding to the average radius of the Rydberg electron. All the parent ions are known to be quite stable, although the geometries of H2F+and CH5+have not yet been fully determined experimentally. Our finding that H30+has CaUsymmetry is of particular interest. Early calculations using Gaussian orbitals16 obtained planar structures (HOH angle = 120O). A subsequent calculation by Diercksen et al.,17which employed a larger Gaussian basis set, predicted a pyramidal structure with an HOH angle of 112-114', in good agreement with recent NMR results.'* Even though we used a minimum basis set for the geometry optimization, we also predict a pyramidal structure with a somewhat larger angle (116') and with an OH bond length in good agreement with Diercksen's calculated value of 1.81-1.84 Q. For CH5+we did not try to determine the geometry (which requires optimizing several parameters) but rather adopted the optimum C, geometry found by Dyczmons et al.lQa In this, the threefold axis of the CH3subgroup coincides with the twofold axis of the Dyczmons CH, subgroup. In a subsequent calculati~n,'~~ and Kutzelnigg found that inclusion of electron correlation led to equivalent energies for the optimized C, and CZu geometries. We chose to use the C, geometry since our basis set gave a lower energy for it than for the CZugeometry. Collation of Energy Levels. Figure 3 plots the electronic energy levels of the second-row hydride molecules and the corresponding levels of the Na and H atoms. Trends in the energy levels correlate with changes in the orbital radii. For Na NeH, the elongation of the core charge distribution induces a substantial lowering of the s-type levels (lower panel), accompanied by a substantial decrease in the orbital radius, whereas the p- and d-type levels (middle and upper panels) move upward and the radii move outward. As the core sprouts more hydrogen atoms and thus in effect becomes more spherical, these levels tend toward the H atom levels, a tendency that becomes more pro-
-
(16)Newton, M.0.; Ehrenson, S. J. Am. Chem. SOC.1971,93,4971, and references therein. (17)Diercksen, G.H. F.; Kraemer, W. P.; Roos, B.0. Theor. Chim. Acta 1975,36,249. (18)Symons, M. C. R. J. Am. Chem. SOC.1980,102,3982. (19)(a) Dyczmons, V.;Staemmler, V.; Kutzelnigg, W. Chem. Phys. Lett. 1970,5,361.(b) Dyczmons, V.;Kutzelnigg, W. Theor. Chim. Acta 1974,33,239.
The Journal of phvsical Chemistry, Vol. 86, No. 18, 1982 9595
Rydberg States of H,, NeH, H,F, H,O, NH,, and CH,
o,wt ln
>
0 K
I.-.
H,O
NH,
CH,
A,
A,
A,
A'
,
..,-.
...-.
-.
...-
.-..'
*.It -0.16'
3*
-0.2
'
H s
IS
I
-
30
-
2s
-
__....-
W
2
H,F
If
..._,..._....-."
5.-.
!
c
NeH
S
-0.06
0)
Y
Na
c
-
..
:
'A
-0.7.6. -0.5
L X+H(,,:l)X=
3
0.00 I'
Q,
?! c
----
- -
c
H,
-
- - -
-
Ne
HF
NH,
CH,
H,O
Flgurr 4. Relatlve stabilbs of ground-state molecules and ions of H,, NeH, H p , H,O, NH,, and CH,. Proton affinities are from ref 3, 17, 19, and 20.
4PJP-
4 -
Na
2. * 0 1
4 -
::I z
-0.1
I
NeH
I
Fl@m9. correletkn of energy levels of Na and H atoms with Rydberg states of NeH (C-v), H,F (C2v),H,O (Csv),NH, (Td),and CH, CC,) molecules. Energies are given relative tothe ionized atom or molecule. The three panels show states correlating with S, P, and D states of the atoms, respecthrety. I n each case, the symmetry species of the molecular levels are indicated.
nounced with increase in the principal quantum number or orbital angular momentum of the corresponding united atom levels. Table 111displays these trends quantitatively, in terms of the quantum defeds. Note that the "symmetry exchangen seen for H2 and H3 (as in Figure 1 and Table 11) does not appear for the second-row hydride molecules. In each case, the ground electronic state correlates with the 3s united atom level and is unstable with respect to dissociation to form X = Ne, -, CHI and an H atom. The exoergicity AE for this dissociation is given by hE = E(X+H) - E(XH) = IP(H) - IP(XH) - PA(X) where PA denotes proton affinity and IP ionization potential of the the ground state. These quantitiea are shown in Figure 4, which was constructed with literature values for the proton a f f i n i t i e ~ ~ J ' Jand ~ * ~our results for the ionization potentials (corresponding to da in Table 111). For NH, the dissociation exoergicity is quite small (AE 31 kJ/mol); indeed, it remains an open question whether the ground electronic state might be stable.21a Our Figure
-
(20) Bondybey, V.; Pearson, P. K.; Schaefer, H. F. J . Chem. Phys. 1972,57, 1123. Dierckeen, G. H. F.; von Nieesen, W.; Kraemer, W. P. Theor. Chim. Acta 1973,31,205. Eadee, R. A.; Scanlon, K.; Ellenberger, M. R.; Dison, D. A.; Mnrynick, D. S. J. Phys. Chem. 1980, 84, 2840. (21) Wan, J. K. 6. J. Chem. Educ. 1968,45,40and referenceatherein. (22) Bishop, D. M.J. Chem. Phys. 1968,48,5285. Schwarz, W. H., E. Chem. Phys. 1975,11,217.
2 -
..... .,. I.
I
4000
!.
1:
I.
6000
I
8000
I
10000
A (A) Flgurr 5. Predicted spectra for transitions between bound Rydberg levels of the wcond-row hydride molecules. For comparison, Na atom transitions are included.
4 and the best theoretical work to date16* predict that the
ground state is unstable but do not rule out the possibility that it is metastable (if a barrier to dissociation exists). Since there is some evidence that NH4 has been detected experimentally$21*23 it appears that the ground state is at least metastable. A detailed theoretical calculation of the NH4 ground-state surface or direct experimental measurements of its lifetime will be necessary to answer the question of its stability. Also indicated in Figure 4 is the asymptote for X + H(n=2) corresponding to the lowest excited electronic state of the dissociation products. Except for NeH, this lies above the ionization limit corresponding to XH+ + e-. Thus, except for NeH, only the repulsive ground-state correlating with X + H(n=l) can cause predissociation of (23) Williams, B. W.; Porter, R. F. J. Chem. Phys. 1980, 73, 5598.
9598
The Journal of Physical Chemistry, Vol. 86, NO. 18, 1982
Raynor and Herschbach
TABLE 111: Quantum Defects of Rydberg Levelsa
H3O
Na
NeH
HlF
S
z+
3s 4s 5s
1.3729 1.3571 1.3527
1.5647 1.5750 1.5513
AI 1.4253 1.4331 1.4742
A, 1.2959 1.2255 1.3875
A1 1.1654 1.0778 0.9052
3P
0.8829
0.8127 0.8418
AI, E 0.6332 0.7993
F, 0.6960
4P
0.8669
0.7388 0.7662
AI, B,, Bl 0.7161 0.7267 0.9958 0.6355 0.6485 0.8652
0.5827 0.6703
0.6235
D
A , TI, E '
3d
0.0103
-0.0485 -0.0345 -0.0208
A,, E, E -0.0581 -0.1204 0.1492
E, F, 0.0364 0.1 224
4d
0.0123
-0.0936 -0.0708 -0.0568
-0.0393 0.0277 0.2397
0.0333 0.1810
P
TI,
x+
B,; AI,,,,; Bl -0.0821 -0.0822 0.0128 0.0452 0.0731 -0.1602 -0.1600 -0.0264 0.0103 0.0890
NH,
CH, A' 1.1510 0.9757 0.7546 A", A', A' 0.5883 0.6402 0.6806 0.5449 0.5705 0.6107 A", A', A", A', A' -0.0006 0.0420 0.0956 0.2035 0.2663 - 0.0035 0.0406 0.1300 0.2164 0.2974
a The energy levels are given in hartree units by Ek = - l / , ( n - dk)-', with d k the tabulated quantum defect. Zero of energy corresponds t o electron plus parent ion a t infinite separation. Symmetries of levels in order of increasing stability are indicated at t o p of each column.
TABLE IV: Predicted Emission Spectra for Second-Row Hydrides molecule
lower and upper levels, transition frequencies, and emission ratesa ( 2 s ) l A., '-~; ( 2 ~ ) 1 A , " ;1E" (3d)3E', (3p)2A1",.E'; (3d)A1',E", (3s)A1';(2p)lA,", ( 2 s ) l A 1 ' A: 5759, 5767,7051; 6 2 9 5 , 6 3 2 3 , 6 5 5 4 ; 6460, 7302 A i f : 0.01,1.55, 4 . 5 4 ; 4 . 4 0 , 3 . 2 3 , 0 . 2 7 ; 0 . 0 , 3 . 2 1 Ef: Ei:
-
I
Ef: E,:
(3p)lTI; 22'; (4s)32+;( 3 s ) l z + (5d)3A, (6s)92+,(4d)2A, 4TI, ( 3 d ) 2 n , 4 2 * ; ( 4 d ) 4 n , 7 2 + ,( 3 d ) 2 n , 3 n ; (4p)3II; (4s)3X', ( 3 p ) l n , 2X+ 5345, 5750,6105, 6129, 8 9 8 6 , 9 0 7 5 , 9 1 6 5 ; 5904, 5920,8589, 8670; 11987; 2890, 3297, 3366 A: A i f : 0.79, 0.96, 1.73, 0.61, 4.57, 1.84, 1.00;0.80, 0.88, 1.98, 2.23;0.97;3.87, 12.7, 6.19 Ef: Ei :
( 3 p ) l B 1 ;l B , ; 2 A 1 ; ( 3 s ) l A 1 (5d)3A2,12A1, 6B1,( 6 s ) l l A , , (4d)2A2,8A,, 4B1, (5s)7A,, ( 3 d ) l A 2 , 4A,, 2B1, (4s)3A1;( 6 s ) l l A , , (4d)9A1, 4B,, (3d)2B2, 5A,, 1 A 2 ;(5d)13A1, ( 6 s ) l l A 1 , (4d)4Bl, 9A,, (3d)2B1, 5A,, 2B1;(3p)2A,, lB,,l B , A: 4353, 4365, 4392, 4 5 5 3 , 4 8 6 6 , 4 8 9 8 , 4 9 6 4 , 5 4 0 8 , 6 6 5 7 , 6 7 8 0 , 6 8 9 2 , 9 3 7 7 ; 6 2 9 8 , 6 7 1 4 , 6 7 1 5 , 1 0 3 2 8 , 10329,11190; 5911, 6377,6804,6804,10541,10542,12153;4307,4344, 5904 Aif: 0.89, 0.84, 0.72, 0.57, 1.68, 2.31, 0.49, 0.69,3.40, 4.43, 1.63, 0.74;0.68, 0.66, 0.64, 1.48, 1.54, 1.44; 0.50, 0.57, 0.69, 0.68, 0.68, 1.59, 1.11;9.32, 9.12, 6.64 Ef: Ei:
( 3 p ) l E ; 2A,; ( 3 s ) l A , (6d)12E, 11E, (7s)12AI, (5d)lOE, 9E, 8E, (6s)9A1,(4d)7A1,6E, 5E, (5s)6A1, (3d)4A1, 3E, 2E, (4s)3A1; (6d)12A1, 12E, (5d)10A1,(6s)9A1,(3d)4A1, 3E, (4s)3A1;(3p)2A1, 1 E A: 5107, 5168, 5215, 5454, 5489, 5613, 5714,6276, 6 3 6 8 , 6 7 1 1 , 7 0 1 8 , 9 1 5 4 , 9 4 0 8 , 1 0 9 2 3 , 1 1 9 0 1 ; 6031, 6056, 6550, 6 9 2 9 , 1 2 7 3 0 , 1 3 2 2 6 , 1 8 7 4 7 ; 5495,6609 Aif: 0.75, 1.84, 0.72, 0.55, 0.69, 0.69, 2.63, 1.85, 1.20, 0.59, 1.90, 1.68, 1.96, 2.70, 1.53;0.89,0.72, 0.51, 0.98, 1.02, 0.97, 0.58;6.48, 5.64 Ef: Ei:
( 3 p ) l F 2 ;( 3 s ) l A 1 (6d)4E, 8 F 2 ,(6s)4A1, (5s)3A1,(4d)2E, ( 3 d ) l E , (4s)2A1,( 3 d ) 2 F 2 ;( 3 p ) l F , A: 5685, 5 7 3 7 , 6 0 8 1 , 7 0 7 8 , 7 3 0 0 , 1 2 2 2 8 , 1 2 7 8 5 , 1 3 4 7 7 ; 8380 Aif: 1.26, 1.46, 2.50, 2.64, 0.63, 2.58, 2.33, 1.89; 3.62 E f : (3p)2A'; 3A'; 1A"; (3s)lA' Ei: (3d)3A"; (5s)13A'; (4s)7A', (3d)6A'; (3d)3A", 6A'; (4d)11A', (3d)5A', (3p)lA", 3A', 2A' h: 12179; 7343,12972, 13957; 1 4 9 7 3 , 1 5 8 1 0 ; 4093, 5535, 7558, 8070, 8547 A s : 0 . 6 3 ; 0 . 5 4 , 0 . 9 7 , 0 . 5 0 ; 0 . 7 2 , 0 . 5 9 ; 0 . 7 6 ,1.02, 3.66,2.85,2.80 a E f gives the levels t o which emission transitions occur; Ei gives the initial levels for the transitions; k is the wavelength in angstroms; and A s is the Einstein emission coefficient for the transition, in l o 7 s - l . The united atom limits for each level are given in parentheses before the level number and symmetry. Where omitted, the united atom limit is the same as the last previously given value. Semicolons are used t o separate transitions t o different final levels; i.e., for NeH, the first six levels listed under Ei, the first six values of h , and the first six values of A s are for transitions from the Ei levels t o the ( 3 p ) l n level; the next four of each are for transitions t o the ( 3 p ) 2 x +level; etc.
any of the excited Rydberg states of XH (shown in Figure 3).
Discrete Emission Spectra. Figure 5 shows the major Rydberg transitions predicted for the hydride molecules.
For comparison, Na atom transitions are included. Table IV lists the lower and upper levels, transition frequencies, and emission rates for all bands with Aif > 0.5 X lo7 s-l. The spectral bands tend to cluster in two regions:
Rydberg States of H,, NeH, H,F, H,O, NH,, and CH,
The Journal of mysical ChemWy, Vol. 86,No. 18, 1982 3507
TABLE V: Electronic States Subject t o Predissociationa molecule
~ O U D
Ne H
c,
H*F H3O
Czv
NH, CH,
c Td c,
3v
VP
GS
RP
B+
B+
n
A, Ai
Ai, Bi Ai,E AI, E, F, A’, A”
A,, B l , A,, E
2;
FSSO (this calcd)
B,
F? A , A“
a Column headed “GS” lists symmetry species of ground electronic state; T P ” indicates states coupled t o ground state by vibronic interaction; “RP” indicates states coupled to ground state by rotation.
500(+7000 A and 9000-12 000 A. All the major bands in the visible correspond in the united atom limit to transitions with 3p (or less often 3s or 3d) as the lower level, but there is a wide range in the upper levels. In the series Na NeH CH6,the spectra tend to shift to longer wavelengths. This reflects the gradual expansion of the mean electronic radii seen in Figure 2. From our error estimates for the energy levels, we expect the band positions to be accurate within a few hundred angstrom units. However, some of the transitions with large calculated emission rates may in fact be quite weak if the upper level is severely predissociated or is not significantly populated during the electron capture process which forms the Rydberg molecule. In addition to the wavelengths the intensities, the line widths of the Rydberg transitions are important characteristics. The transition width is governed primarily by coupling of the lower level to the dissociative ground electronic state. This had a key role in the discovery of H3, as Herzberg found’$ very broad lines for the transitions with (2s)A,’ as the lower level and much narrower lines for transitions with (2p)Afl as the lower level. These levels are subject, respectively to vibration-induced and to rotation-induced predissociation by the repulsive (2p)E’ ground state. For the second-row hydride molecules, the selection rules for predissociation” are particularly simple. Since the dissociative ground state is totally symmetric, the vibration- or rotation-induced couplings occur for excited electronic states with symmetry species corresponding to those of vibrational or rotationl modes, respectively. Table V lists the states subject to predissociation. In analogy to the H3 case, for each symmetry species predissociation via the ground state can be expected to be most prominent for the lowest state of that species. Predissociation of the higher states is likely to be inhibited by unfavorable Franck-Condon factom2 Thus, emissions to the lowest states with symmetry species listed under “VP”in Table V are expected to have quite broad lines, whereas emissions to states that appear only under “RP” will have much narrower lines that gradually widen with increasing rotational quantum numbers. Continuous Emission Spectra. Table IV includes some predicted transitions to the dissociative ground state (correlating with the 3s united atom level). Only the most intense transitions falling in the visible region are listed. As noted above, the NH, ground state is probably at least metastable. There is some e ~ p e r i m e n t aand l ~ ~theoretica125*26 evidence for the existence of a metastable H 3 0 ground state. Niblaeus et ala,%in a CI calculation, found a dissociation barrier of -12 kJ/mol, but their calculated zero-point energy for the molecule was -16 kJ/mol.
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(24) Wight, G.R.;Brion, C. E. Chem. Phys. Lett. 1974,26,607, and referencestherein. (25) Cangi, R. A.; Bader, R. F. W. Chem. Phys. Lett. 1971, 11, 216. Siegbahn, P. E. M. Chem. Phys. (26) Niblaeus,K. S. E.; Roos, B. 0.; 1977,25, 207.
TABLE VI: Experimental and Theoretical Spectra for HP
transition (3p)A,” + (2s)A,’ (3p)E‘ -+ (2s)A,‘ (3d)A1‘ ( 2p)A,” (3d)E” + (2p)A,” (3s)A1’-+ (2p)A,” -+
exptb A A 5600 5767 7100 7051 6295 5900 6323 6025 6554
ref 3c
Aij
A
Aij
1.55 4.54 4.40 3.23 0.27
5680 6980 6320 6350 6620
2.27 4.62 4.34 3.14 0.45
A is the wavelength in angstroms and Au is the Einstein emission coefficient for the transition, in units of lo’ s-’. From ref 1 and 2. The wavelengths given here are from Table XI11 of ref 3 and are corrected for geometry and zero-point energy.
However, they also found that the barrier height was very sensitive to the basis set used and a more complete basis set might therefore yield a stable ground vibrational state. The existence of a metastable H2F or NeH ground state is much more unlikely. H3.Our calculations (Table 11)serve to demonstrate the efficacy of the FSSO approach, which appears practically The equivalent to the procedure of King and M~rokuma.~ transition wavelengths we find (Tables IV and VI) typically agree within -70 A and the emission rates within 10%. Agreement with the observed spectra2 (Table VI) is satisfactory (within 50-150 A) for the transitions near 5600 (%)A[ and (3p)E’ and 7100 A, assigned as (3p)A; (2s)A1’. Agreement is less satisfactory (only within -500 A) for the observed bands near 5900 and 6025 A, assigned by Herzberg as (3d)E” (2p)Afl and (3s)Ai (2p)A;. According to the calculated emission strengths, the latter transition should be quite weak: tenfold weaker than the (3d)A,’ (2p)A; transition in the same spectral region. That transition does not offer a tenable reassignment, however, because it should occur at shorter wavelengths than the (3d)E” (2p)Afl transition. Examination of our diatomics-in-molecules (DIM) calculations12for these transitions appeared to offer an explanation of this discrepancy. The DIM calculations gave agreement with the experimental assignments for transitions to the (2p)A/ level. This suggested that the FSSO and Gaussian3 calculations were inadequate due to breakdown of the Lfrozen core approximation”, as in the notorious case of polyene ~ p e c t r a . ~ ’However, ?~~ CI calculationsmusing the same FSSO wave functions reported here gave predictions identical with those presented in Table IV and VI for H3 except that the transition wavelengths were shorter by 100 A or less. Thus we do not yet understand the source of the discrepancy between theory and experiment. NeH. The strongest predicted transitions correspond 3p in the united atom limit and appear near to 3d -9OOO A. Since that region is less amenable experimentally, the several weaker transitions near -6000 A may be of more interest. The strongest among these, 4d A 3p II,is predicted to appear near 6100 A and should exhibit relatively narrow line widths because predissociation of the lower state is vibronically forbidden. H$. The most promising transitions for identification of this molecule are five strong transitions, all with (3p)B1 as the lower state. These comprise a pair of transitions near 4900 A with upper states arising from 4d in the united atom and a triad of transitions near 6800 A with upper
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(27) Hudson, B. S.; Kohler, B. E. Chem. Phys. Lett. 1972, 14, 299. (28) Schulten, K.; Karplua, M. Chem. Phys. Lett. 1972, 14, 305. (29) Raynor, S.;Herschbach, D. R.Manuscript in preparation.
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The Journal of phvsical Chemistry, Vol. 86, No. 18, 1982
states arising from 3d. All these transitions are likely to be quite broad, since predissociation of the lower state (3p)B1 by the ground state is vibronically allowed. The initial search thus should use D2F in order to reduce the broadening by predissociation. Also of intereat are some considerably weaker transitions near 6500 A with (3p)Bz as the lower state; these should give narrow lines since predissociation of (3p)B2by the ground state is vibronically forbidden. H30.Again there are five most promising transitions for identification, all with (3p)E as the lower state. These transitions appear near 5200,5700,6300,6400, and 7000 A with (6d)E, (6s)A, (4d)A, (4d)E, and (5s)A as upper states. Jahn-Teller distortions may affect all the degenerate states, including (3p)E. Again the search should use the deuterated molecule since predissociation of the lower state (3p)E by the ground state is vibronically allowed. The weaker transitions near 6000,6500, and 7000 A with (3p)A1 as the lower state may be observable but since predissociation of (3p)A1 is likewise vibronically allowed these will also be broad. If the ground (3s)A1state proves to be metastable with a substantial lifetime, the spectrum should exhibit two prominent transitions which have very large emission coefficients. These come from (3p)A1and (3p)E and occur near 5500 and 6600 A, respectively. NH,. Most of the transitions of interest have (3p)F2as the lower state, for which predissociation by the ground state is vibronically allowed. Two strong transitions are predicted near 5700 A with (6d)E and (6d)Fz as upper states. These are subject to Jahn-Teller distortion, as is (3p)F,. Two even stronger transitions are predicted near 6100 and 7100 A with (6s)Ai and (5s)A1as upper states. The bands Henberg has identified6for NH4 appear at 5690 A (Schuster band, wrongly attributed to ammonia for over a centUry30) and as a close doublet at 6635,6637 A (Schuler band, discovered but not attributed 25 years ago31). Our results indicate that two other strong band systems should be found. The most likely assignment for the Schuler band is (5s)A1or &)Al (3p)Fz,in agreement with Herzberg's assignment of (3p)F2as the lower level. For the Schuster band there are three plasuible assignments, (6s)Ai, (6d)E, or (6d)Fz (3p)Fz. However, Herzberg has assigned this transition to (3d)F2 (3s)A1on the basis of linewidth and Coriolis effects. This indicates that a barrier to dissociation exists in the ground (3s)A1state of NHI. Our calculations predict the (3d)F2 (3s)A1 transition would have an emission coefficient of only 0.02 X lo' s-l and appear at -5170 k As with H,, the source of this large discrepancy remains an open question.
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(30) Schwter, A. Rep. Brit. Assoc. 1872,38. (31) Schuler, H.;Michel, A.; Grun, A. E. 2.Naturforsch. A 1955, IO, 1.
Raynor and Herschbach
As previously mentioned, we recalculated the energy levels for NH, with the NH bond lengths varied by h0.05 bohr to test the effect of geometry on the predicted energies of the levels. This calculation was done with a smaller Rydberg basis but included vibrational frequencies, potential minima, and Franck-Condon overlaps for the transitions between Rydberg levels. We found that the vibrational frequencies varied only from 3500 to 3670 cm-' and the optimum NH bond lengths varied by no more than fO.O1 bohr from the nominal geometry. Thus, all of the Franck-Condon overlaps for Au = 0 transitions were equal to unity within three significant figures and no AIJ # 0 transitions appear likely. We also evaluated the differences between wavelengths for transitions from u = 0 in the upper state to u = 0 in the lower state and transitions between the potential minima for the nominal geometries. The largest difference was 130 A for the (4s)A (3p)F2 transition and the other differences were -50 A or less. In all cases, the wavelengths for the 0 0 transitions were shorter than those obtained for the nominal geometry. We have not yet investigated the effects of Jahn-Teller distortions on the predicted wavelengths. CH,. For this molecule, the time required for a full FSSO + Rydberg calculation was prohibitive. We therefore tried a drastically simplified treatment, with a subminimum basis set (SMBS). Orbitals on the hydrogen atoms were omitted altogether; Is, 2s, and 2p orbitals were placed on the carbon atom of the parent ion and the exponents optimized. The same basis set of 3s,3p, and 3d orbitals used for the other molecules was then added to simulate the CH5 Rydberg levels. As a test, we applied the same procedure to NH, and found that the orbital energies obtained from the SMBS calculation were higher than from the FSSO + Rydberg calculation; the worst error was 3 X loh3hartrees and occurred for the dissociative ground (3s)A1state. For the bound Rydberg states, the largest error was 2 X hartrees for the (3p)F2 state. This performance by the SMBS is surprisingly good and serves to emphasize the resemblance of these molecules to perturbed H atoms. The excellent agreement between our results and the valence bond treatment of Strehl et al.14for NH4 offers further support for this approximation. In Figure 5 we did not include CH5 because only one transition with As > 0.5 X lo7s-l appears in the wavelength (3p)A' transition, region shown. This is the (5s)A' predicted at -7300 A (with a large uncertainty, estimated as i700 A); it should be strongly broadened by predissociation.
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Acknowledgment. We thank G. Herzberg and H. F. King for enjoyable and resonant discussions. We are grateful for support of this work to the Department of Energy under Contract DE-AC02-79ER10740.
