J. Phys. Chem. 1988, 92, 1821-1830
1821
of the method to represent the D = 3 system thus does not depend particle problems. Again, techniques based on Monte Carlo on the magnitude of the interaction but only on its dimension methods are applicable to tunneling in many degrees of freedom.32 dependence. Our results for the hydride ion add to the evidenThe multiple minima introduced by symmetry breaking are that major features of the electron correlation are included technically challenging but heuristically encouraging. As seen in both D 1 and D m limits, even in the lowest order apin Figure 5 , for H- these features suggest that something like shell proximations. In particular, much of the effect of short-range structure can appear even in the classical limit. The Lewis structure then indicates directly how the electrons are distributed electron repulsion apparently is contained in the residues of the among orbitals. A similar a n a l y s i ~ ~of~ the , ' ~ large-D limit for D = 1 poles. New methods are required if we aim for higher accuracy in the Li atomi4 finds a kindred symmetry breaking occurs for 2 applying dimensional continuation to strongly correlated and < 2.29, with two equivalent inner electrons and a distinct outer weakly bound systems such as the hydride ion. For this purpose, electron. Analogous effects appear at large D for molecules when extending the perturbative approach appears less promising than either the nuclear charges or the internuclear distances are varied. efficient semiclassical procedures for obtaining vibrational eiThus, for the H2+and H2 molecules,34 the effective potential genstates using basis sets derived from classical trajectories.25 localizes the electron(s) in the plane bisecting the internuclear Direct semiclassical quantization may also prove f e a ~ i b l e . ~ ~ - ~ *axis when the separation between the nuclei is small but the These methods should be readily applicable for large D, where potential acquires a double minimum when the separation between the effective potential becomes nearly harmonic. There are three m limit hence builds the nuclei becomes large enough. The D into the zeroth-order model both chemical bonding and the proper chief issues: (1) Practical semiclassical techniques need to be developed for systems with many degrees of freedom. Monte Carlo dissociation behavior. Such features suggest the likely utility of methods now offer a promising approach.29 (2) When the efthe large-D regime in developing semiclassical interpretations of electronic structure. fective potential has no minimum in the large-D limit, scattering theory techniques may be needed to treat correlation effects. (3) Acknowledgment. We have enjoyed discussing many dimenWhen the effective potential has two or more equivalent or comsions of this study with John Loeser, David Goodson, and Don parable minima, tunneling among them must be accurately Frantz. We also thank Rudy Marcus for his incisive contributions evaluated. Pertinent approaches include supersymmetry techand exuberant advocacy of semiclassical methods which have niques recently applied to large-D expansions30and to semiclassical strongly influenced our work. t ~ n n e l i n g ,but ~ ' so far these methods deal just with simple one-
-
-
-
Registry No. H-, 12184-88-2. (25) Frederick, J. H.; Heller, E. J. J . Chem. Phys. 1987, 87, 6592. (26) Strand, M. P.; Reinhardt, W. P. J . Chem. Phys. 1979, 70, 3812. (27) Maslov, V. P.; Fedoriuk, M. V. Semiclassical Approximations in Quantum Mechanics; Reidel: Boston, MA, 1981. (28) Delos, J. B. Adu. Chem. Phys. 1986,65, 161. Knudson, S. K.; Delos, J. B.; Nord, D. W. J . Chem. Phys. 1986,84, 6886. (29) See several papers in: J . Stat. Phys. 1986, 43, 729-1237. For an application to H-, see: McDowell, H. K.; Doll, J. D. Chem. Phys. Lett. 1981, 82, 127. (30) Imbo, T. D.; Sukhatme, U. P. Phys. Reu. Lett. 1985, 54, 2184.
(31) Bernstein, M.; Brown, L. S. Phys. Rev. Lett. 1984,52, 1933. Kumar, P.; Ruiz-Altaba, M.; Thomas, B. S . Ibid. 1986, 57, 2749. Keung, W. Y.; Kovacs, E.; Sukhatme, U. P. Ibid., in press. (32) Garett, B. C.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 4931. Doll, J. D.; Coalson, R. D.; Freeman, D. L. Ibid. 1987, 87, 1641. Chang, J.; Miller, W. H. Ibid. 1987, 87, 1648. (33) Loeser, J. G., private communication. (34) Frantz, D. D.; Herschbach, D. R., to be submitted for publication.
Rotational-Vibrational Structure of a Quasi-Linear Molecule: CH,' Jae Shin Lee and Don Secrest* School of Chemical Sciences, University of Illinois, Urbana, Illinois 61 801 (Received: August 17, 1987)
A new potential energy function is obtained for the ground electronic state of CH2+through a Simons-Parr-Finlan (SPF) type expansion of an ab initio potential surface for this molecule. The SPF type potential is found to fit the a priori potential points extremely well and has reasonable physical properties along the vibrational coordinates of the molecule. The rotation-vibration states of this molecule are calculated for J = 0, 1,2, and 3 by using this potential function. The calculations were carried out using a linear molecule Hamiltonian. Assignments have been made to each vibrational state. It is possible to identify the (1 1 0) state through the rotational structure of the molecule. This level was assigned to (0 4 0) previously. As in the previous calculations on this molecule, the Renner-Teller effect was neglected.
Introduction Floppy molecules may undergo rather large excursions during normal vibrations. There have been a number of studies in the recent literature on molecules of this type.'J In particular, molecules with a low barrier to linearity are of interest because of the singularity in the Hamiltonian for the linear configuration. This has been handled by Carter and Handy3 by using a linear molecule Hamiltonian, and by Tennyson and S ~ t c l i f f eusing ~,~ (1) Bacic, Z.; Light, J. C. J . Chem. Phys. 1986, 85, 4594. (2) Bowman, J. M. Arc. Chem. Res. 1986, 19, 202. (3) Carter, S.; Handy, N. C. J. Mol. Spectrosc. 1982, 95, 9.
0022-3654/88/2092-1821$01,50/0
Jacobi coordinates and a technique related to close coupling techniques in scattering theory. Both of these calculations were performed on the CH2+ molecule which has a low barrier to linearity, using a potential computed by Carter and Handy.3 This potential is a sum of diatomic potentials with a three-particle term obtained by a fit to an ab initio calculation of Bartholomae, Martin, and Sutcliffe.6 (4) Tennyson, J.; Sutcliffe, B. T. J . Mol. Spectrosc. 1983, 101, 71. (5) Tennyson, J.; Sutcliffe, B. T. J . Chem. Phys. 1982, 77, 4061; 1983, 79, 43. (6) Bartholomae, R.; Martin, D.; Sutcliffe, B. T. J. Mol. Spectrosc. 1981, 87, 367.
0 1988 American Chemical Society
Lee and Secrest
1822 The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 TABLE I: i i k 0 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 2 2 0 3 2 0 4 2 0 3 3 0 0 0 1 ~~
SPF Expansion Coefficients (in hartrees) d;,i
~
1 0 1
2 0 3 0 4 0
1
1
1
1
I I
2 1 1 3 1 1 2 2 1 0 0 2 1 0 2 2 0 2 1 1 2 0 0 3
0.2601820 (-2) 0.7834739 (0) -0.453 1045 (0) -0.1596746 (0) 5.1018130 (0) 4.7462950 (0) -6.0224740 (-2) 0.648 209 2 (0) 5.4890890 (0) -3.4816230 ( I ) -5.0877190 ( I ) -9.406 163 0 (0) 2.1427310 ( I ) 4.7420520 (2) -2.5065720 (2) -0.227601 6 (-1) -0.2183118(-1) -0.651 5987 (-1) 0.211 7776 (0) 0.7737005 (0) 0.1422707 (0) 0.3932465 (-1) -6.4706980 (0) 3.3658730 (0) 0.5280461 (-1) -0.1517588 (-1) -0.311681 3 (-1) -0.4710155 (-1) -0.121 1573 ( - I )
equilib bond distance R, = 1.094 A
iltonian for a bent molecule. The molecule can easily achieve a linear configuration in normal vibration, and in the present case the zero-point vibrational energy is above the barrier to linearity. The general formula defining the transformation from spacefixed Cartesian coordinates to internal coordinates for an N-atom molecule is
+ S'(c@y)(r: + &,)I')
r, = R equilib bond angle 8, = 139.8'
saddle point for the linear molecule
ro = 1.089577 A see eq 13 and 14
The use of the linear molecule Hamiltonian as described by Carter and Handy lead to some convergence difficulties and they were only able to compute band origins. Tennyson and Sutcliffe were able to compute the J = 0 and J = 1 states. They indicated that the calculation of the J = 1 state was expensive and did not extend the calculation to higher rotational states. We have recently developed a linear molecule Hamiltonian' which is ideally suited to the computation of rotation-vibration energy levels of floppy molecules.s This method has been applied to the study of a van der Waals molecule but it has not as yet been applied to a problem that had been studied by other techniques. Furthermore, we have found from previous experience that our technique handles the rotation-vibration problem so efficiently that we would have no difficulty studying the rotation-vibration spectrum of the CH2+molecule. In our previous study of a van der Waals molecule we found a method of fitting potentials developed by Simons, Parr, and Finlan9 (SPF) to be useful. In the present paper we refit the original data of Bartholomae, Martin, and Sutcliffe6 by a method related to the S P F fitting method and compare the results of the rotation-vibration energy level calculations using both our S P F fit and the Carter and Handy fit.
( n = 1, 2, ..., N ) (1)
Here R is the center of mass vector of the system, r, is the Cartesian position vector of the nth atom, and r: is the constant vector corresponding to the equilibrium position of the nth atom in the molecule-fixed system. S(afly) is a 3 X 3 matrix which rotates the space-fixed coordinate system by the Euler angles, a, fl, and y into the molecule-fixed coordinate system, and /(") is the 3 X ( 3 N - 6) matrix whose element, $1, determines the amplitude of the displacement which a particular vibrational coordinate 7, contributes along the q molecule-fixed axis for the nth atom. Thus the choice of internal coordinates is a choice of I("). For a linear symmetric triatomic molecule ABA, our choice of internal coordinates are
vu+-+-1 Here, C represents another end atom A, and vs, qa, q b are the vibrational coordinates which represent the symmetric stretch, asymmetric stretch, and bend mode of the molecule, respectively. The constant ro is the position of the minimum in the potential when the molecule is held linear. It is the z coordinate of the saddle point in the barrier to linearity for this molecule. In this internal coordinates system, the long axis of the molecule is oriented along the (molecule-fixed) z axis and all (internal) motions of the molecule are confined to the xz plane of the molecule-fixed system. Using these coordinates, the rotation-vibration Hamiltonian of a linear symmetric triatomic molecule is written as H=
1
a2
mA
Theoretical Method
mA
a2
mBM dTa2
mA
mBM c
(
__ d2 dl)b2
+ I d + 'Ib a ' I b /
The rotation-vibration Hamiltonian operator for a polyatomic molecule was derived' by direct coordinate transformation of the quantum mechanical Hamiltonian expression from Cartesian coordinates to internal coordinates, using the methods of tensor analysis as developed by Podolsky.Io One of the advantages of this approach is the ability to use various internal coordinate systems which would give the better convergence when the variation method is used for solving the rotation-vibration problem of a molecule." For a floppy molecule like CH2+with a small barrier to linearity, Carter and Handy3 have shown that a Hamiltonian for a linear molecule is more appropriate than a Ham(7) Estes, D.; Secrest, D. Mol. Phys. 1986, 59, 569. (8) Lee, J . S.; Secrest, D. J . C h e m Phys. 1986, 85, 6565. (9) Simons. G.; Parr, R. G.;Finlan, J. M. J . Chem. Phys. 1973, 59, 3229. Brown, J. E.; Parr, R.G. J . Chem. Phys. 1971,54. 3429. Simons. G.J . Chem. Ph>,s. 1972, 56. 4310. (IO) Podolsky, B. Phys. Rec. 1928, 32, 812. ( 1 1 ) Chen. C . L.; Maessen, B.; Wolfsberg. M . J. C h e m Phys. 1985, 83. 1795
(3)
where J , = -i
a + cos y cot + sin y ab
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1823
Structure of a Quasi-Linear Molecule CH2+
TABLE II: Handy-Carter and Simons-Parr-Finlan Fits to the Bartholomae-Martin-Sutcliffe Potentialo TAB rBC 4 deg E(BMS) EWC) AE(HC) 10.14 -10.14 2.09 2.09 138 0.00 1.94 1.77 2.09 1.92 1.78 2.25 2.08 1.94 1.77 2.09 1.92 1.77 2.25 2.08 1.96 1.79 2.1 1 1.94 1.80 2.27 2.10 1.95 1.79 2.11 1.94 1.79 2.27 2.10 2.01 1.85 2.16 1.99 1.86 1.69 2.31 2.15 2.00 1.84 2.15 1.99 1.84 2.31 2.15
2.10 2.26 2.25 2.42 1.93 2.41 2.58 2.09 2.26 2.25 2.41 1.93 2.41 2.58 2.12 2.29 2.27 2.44 1.96 2.43 2.60 2.11 2.27 2.26 2.43 1.94 2.42 2.59 2.17 2.33 2.3 1 2.48 2.01 2.18 2.47 2.64 2.15 2.31 2.30 2.46 1.99 2.46 2.62
144 144 143 143 145 142 142 133 132 132 132 133 132 132 155 155 153 154 157 152 152 121 121 122 122 120 122 122 168 168 165 165 171 171 163 163 109 108 110 110 107 111 111
790.99 6089.19 1043.36 44 16.5 1 5321.76 4265.53 6270.17 964.99 6251.76 1082.14 4440.13 5780.19 4233.58 6224.72 951.44 5859.12 1682.47 4831.38 4475.57 5139.85 7020.84 1732.71 6612.95 2003.12 5 11 1.96 6175.82 5199.55 7043.38 1395.16 5492.42 3016.70 5682.24 3110.94 9416.76 6929.36 8534.18 3706.05 7788.37 4357.38 6974.33 7188.28 7650.73 9205.05
904.64 6358.61 1222.35 4913.12 5917.56 4558.53 683 1.12 947.82 6446.71 1102.18 4850.41 62 18.42 4386.19 6708.90 1029.26 6061.52 1842.84 5233.96 4936.89 5310.57 7391.67 1586.09 6740.34 1854.19 5402.27 6344.62 5083.66 7302.39 1282.23 5506.74 2938.84 5783.10 3206.90 9295.23 6666.07 8409.52 3852.28 8233.13 4376.64 7428.25 7503.50 7400.84 9329.22
-1 13.65 -269.42 -178.99 -496.61 -595.81 -293.00 -560.95 17.17 -194.96 -20.04 -410.28 -438.23 -152.61 -484.18 -77.82 -202.41 -160.37 -402.5 8 -461.32 -170.72 -370.83 146.62 -127.39 148.93 -290.3 1 -168.80 1 15.90 -259.01 112.93 -14.32 77.86 -100.86 -95.96 121.52 263.30 124.66 -146.24 -444.76 -19.26 -453.92 -315.22 249.89 -124.1 7
E(SPF)
AE(SPF)
-2.26 784.92 6084.23 1046.94 4419.16 5324.72 4268.97 6273.87 95 1.48 6239.88 1071.39 4424.26 5774.20 4228.47 6214.49 961.70 5870.85 1700.45 4852.20 4475.58 5145.14 703 1.42 1746.70 6633.09 2009.3 5 5112.21 6181.06 5201.26 7040.82 1392.92 5480.24 3000.17 5673.23 31 11.44 9419.63 6928.64 8529.29 3702.97 7781.98 4356.58 6 979.2 8 7186.74 7649.27 9207.96
2.26 6.06 4.96 -3.58 -2.64 -2.97 -3.43 -3.70 13.50 11.88 10.75 15.87 5.99 5.11 10.23 -10.25 -11.73 -17.98 -20.81 -0.01 -5.29 -10.58 -13.99 -20.14 -6.23 -0.25 -5.24 -1.71 2.56 2.24 12.18 16.53 9.01 -0.49 -2.88 0.72 4.89 3.08 6.39 0.8 1 -4.95 1.53 1.46 -2.91
OBond lengths are in bohr and energies in cm-I. See ref 6 for exact bond lengths and angles.
sin y
Jy = -i
a
h = sora
(11)
where H,,, is the Hermite polynomial and LLm , is the generalized Laguerre polynomial with n = ( n b - I q ) / 2 and m = lq with n b K = even. DMK,(J)(a&)is the rotation matrix element which is an eigenfunction of and J,. We adopt the convention of EdmondsIz in defining the DMK(J),However, due to the symmetry condition imposed on the total wave function by the Pauli exclusion principle, we used linear combinations of the basis function given in eq 12 for the K # 0 cases. The total wave function must be antisymmetric with respect to the interchange of two hydrogen atoms because 'H is a fermion. For this molecule, both the antisymmetric and symmetric basis functions are allowed for the nuclear triplet and singlet, respectively.
+
and
+
= Jx2 Jy2
+ J:
(7)
J+ = J, - iJy
(8)
+ iJy
(9)
J- = J,
Here, J represents the total angular momentum operator of the molecule which acts only on the Euler angles, cy, 0, and y, and J, is its component along the molecule-fixed z axis. The J+ and J- are the raising and lowering operators for the molecule-fixed z component of the angular momentum. V is the molecular potential energy function which depends on the vibrational coordinates only. The wave function is chosen to be an eigenfunction of and its space-fixed z component since they are always good quantum numbers.
GJM= c c i D f ; l ! , ( d y )dn,,,,(qs) I
The basis functions are
4n,(,,(.)ta) 4 n b ( , ) K ( , , ( ? b )
(lo)
A Potential Energy Function for CHz+
An accurate ab initio CI potential energy surface for the ground electronic state of CH2+ molecule has been computed by Bartholomae, Martin and Sutcliffe.6 Following the approach suggested by Simons,13and Simons, Parr, and Finlang (SPF), we have (12) Edmonds,.A. R. Angular Momentum in Quantum Mechanics; Princeton University Press: Princeton, NJ, 1960. (13) Simons,G. J . Chem. Phys. 1974, 61, 369.
1824 The Journal of Physical Chemistry, Vol. 92, No. 7, 1988
Lee and Secrest
F"
0.0
0.4
I O
0.0
I 4
27
' b
'b
F
-
-08
'
co
1
I
,
I
04
IO
I 4
-1.8 -27
0.0
7b
27
'b
Figure 1. Contour plots of the symmetric stretch and bend for va = 0. The S P F potential is plotted in (a, top left) and the H C potential in (b, bottom left). The original points of Bartholomae, Martin, and Sutcliffe6 are projected onto this plot to give a feeling for the region covered by the original points. The SPF(c) and HC(d) potentials are plotted for larger displacements in (c, top right) and (d, bottom right), respectively, to show that while the two potentials differ significantly outside of the well region both behave reasonably in the region not covered by the a priori points. The contours are at 50, 100, 200 cm-I, etc. 07
fit these points using an expansion in the variables p l , p2, and p3, where
Fa 0.0
rBC P2
=
- rO
-0 7
-0 7
0
I05
s' 0.7
p
oc
-0 7 -07
0
I05
In these expressions, A and C represent a hydrogen atom and B is the carbon atom. The ro is the bond length between the carbon and hydrogen atoms at the minimum of the potential when the molecule is held linear, 0 is the bond angle, and rABand rBc are the instantaneous internuclear distances between atoms A and B, and B and C, respectively. For p3, we employed a trigonometric function t o produce the correct symmetry around the linear configuration, rather than using conventional SPF expansion variable, (0 - 0 , ) / 6 . The SPF expansion of potential functions obtained from ab initio calculations or experimental results has been successfully used to calculate the band origins of a number of triatomic molecule^.'^*^^ Our coordinates are somewhat different from the usual SPF coordinates in that they are not displacements from the equilibrium position but rather displacements from the saddle point to linearity. We were able to fit the Bar-
7s
Figure 2. Contour plots of the symmetric stretch against the asymmetric stretch with the molecule held bent in its equilibrium displacement. The SPF potential (a, top) and the H C potential (b, bottom) differ significantly in the region covered by the a priori data points of Bartholome, Martin, and Sutcliffe.6 The points are projected onto the plots.
(14) Carney, 1976. 61. 371.
G.D.; Curtis, L. A,; Langhoff, S . R. J . Mol. Spectrosc.
( 1 5 ) Carney, G.D.; Porter, R. N . J . Chi" Phys. 1976.65, 3547; Chem. Phys. Lett. 1977, 50, 327. (16) Sorbie, K . S.; Murrell, J. N . Mol. Phys. 1975, 29, 1387.
Structure of a Quasi-Linear Molecule CH2+
The Journal of Physical Chemistry, Vol. 92, No, 7, 1988 1825
TABLE 111: Potential Fits Reexpanded in Local Functions Given by Q 17-19'
8, deg
rBC
TAB
2.09 2.10 2.26 2.25 2.42 1.93 2.41 2.58 2.09 2.26 2.25 2.41 1.93 2.41 2.58 2.12 2.29 2.27 2.44 1.96 2.43 2.60 2.11 2.27 2.26 2.43 1.94 2.42 2.59 2.17 2.33 2.31 2.48 2.01 2.18 2.47 2.64 2.15 2.31 2.30 2.46 1.99 2.46 2.62
138 144 144 143 143 145 142 142 133 132 132 132 133 132 132 155 155 153 154 157 152 152 121 121 122 122 120 122 122 168 168 165 165 171 171 163 163 109 108 110 110 107 111 111
2.09 1.94 1.77 2.09 1.92 1.78 2.25 2.08 1.94 1.77 2.09 1.92 1.77 2.25 2.08 1.96 1.79 2.11 1.94 1.80 2.27 2.10 1.95 1.79 2.11 1.94 1.79 2.27 2.10 2.01 1.85 2.16 1.99 1.86 1.69 2.31 2.15 2.00 1.84 2.15 1.99 1.84 2.31 2.15
E(HC) 8.89 904.24 6356.03 1222.13 4915.14 5916.08 4560.25 6835.72' 953.38 6440.17 1101.91 4851.97 6223.66 4386.81 6710.60 1025.20 6064.32 1842.96 5230.37 4938.38 5309.43 7406.40 1586.74 6737.39 1851.94 5398.87 6401.OO 5081.62 7300.89 1285.31 5509.17 2936.55 5778.55 3204.43 9281.50 6665.26 8447.63 3858.87 8149.32 4374.70 7443.71 7398.70 7406.76 9333.26
E(BMS) 0.00 790.99 6089.19 1043.36 4416.51 5321.76 4265.53 6270.17 964.99 6251.76 1082.14 4440.13 5780.19 4233.58 6224.72 951.44 5859.12 1682.47 4831.38 4475.57 5139.85 7020.84 1732.71 6612.95 2003.12 51 11.96 6175.82 5199.55 7043.38 1395.16 5492.42 3016.70 5682.24 3 1 10.94 9416.76 6929.36 8534.18 3706.05 7788.37 4357.38 6974.33 7188.28 7650.73 9205.05
AE(HC) -8.89 -1 13.25 -266.84 -178.77 -498.63 -594.32 -294.72 -565.55 11.60 -188.41 -19.77 -41 1.85 -443.47 -153.23 -485.88 -73.76 -205.20 -160.49 -398.99 -462.81 -169.58 -385.56 145.97 -1 24.45 151.19 -286.91 -225.18 117.93 -257.51 109.84 -16.75 80.15 -96.31 -93.48 135.26 264.10 86.55 -152.82 -360.95 -17.31 -469.38 -210.42 243.97 -128.21
E(SPF) -4.85 803.68 6355.50 999.59 4326.43 5136.24 4252.70 6218.23 987.38 6562.49 1027.25 4362.32 5687.64 4302.97 6282.86 981.91 5858.09 1714.92 4794.03 4257.88 5132.00 6989.44 1754.48 6726.26 1986.33 5034.43 6165.84 5226.90 7160.95 1479.00 5327.53 3038.09 5802.79 2994.66 9355.31 7026.37 8642.57 3696.27 7716.27 4397.09 6923.87 7194.56 7458.03 9311.91
AE(SPF) 4.85 -12.69 -266.30 43.77 90.08 185.52 12.83 51.95 -22.40 -310.73 54.90 77.80 92.56 -69.40 -58.14 -30.47 1.03 -32.45 37.36 217.69 7.85 3 1.40 -21.78 -1 13.31 16.79 77.53 9.98 -27.34 -1 17.57 -83.84 164.89 -21.39 -120.55 116.29 61.45 -97.01 -108.39 9.78 72.09 -39.71 50.46 -6.29 192.71 -1 06.86
Bond lengths are in bohr and energies in cm-'. See ref 6 for exact bond lengths and angles. TABLE I V Converged Quantum Number of Each Vibrational Mode for J = 0, 1, 2, and 3
Handy-Carter potential ns
J=O singlet triplet J=1 singlet even singlet odd triplet even triplet odd J=2 singlet even singlet odd triplet even triplet odd J=3 singlet even singlet odd triplet even triplet odd
nb
SPF potential
na
ns
nb
na
20 16
6 5
9
18
8
16
10 11
17 18 17 18
7 7 6 6
8 8 8
17 18
11 11
18
9
17
10 12
18 19 19
8
18
9 8
17 19
12 10 12
15
6 6 6 6
8
17
10
9
19
7
9 10 12
18 18 18
9 6 9
7 8
19 18
11 11
7 9
18 18
10 11
9
9 9 8 11
9 9 10 10 8
6,"
47
Pb
2300
Pa
7400
Ps
30
Pb
2600
Pa
10700
'6 in A-2. tholomae, Martin, and Sutcliffe potential surface by a sixth-order expansion in these variables with high accuracy.
While the Carter and Handy potential3 was constructed to be reasonably accurate over the whole region of coordinates of triatomic system, thereby reducing the accuracy of fitting the lower energy points, our SPF-like expansion potential was found to fit the lower energy points extremely well without sacrificing the accuracy of higher energy points or changing the potential behavior in these regions significantly. Though the fit always behaves well in the region of the known Bartholomae, Martin, and Sutcliffe points, it initially had an erratic long-range behavior. This is easily corrected by constraining the potential to remain larger at long distances from the potential well. In the present case, this was accomplished by requiring the potential to go through a highenergy point when the end atoms come close together. This physically reasonable constraint gave a potential which looked physically reasonable asymptotically and still fit the known points well. The fit parameters d,,k and the equilibrium geometry are given in Table I. We compare the original data of Bartholomae, Martin, and Sutcliffe in Table I1 with the Carter and Handy potential and the SPF fit that we have produced. The difference of accuracy between the two potentials is very clear. In Figure la, we plot the symmetric stretch versus the bend, with asymmetric stretch la= 0. A double well appears along the direction of bend mode, corresponding to the bent equilibrium geometry of CH2+. Here and in all figures in this paper, the contours start at 50 cm-' and double in frequency successively such that the nth contour is 50 X 2" cm-I. In Figure lb, we plot the Carter and Handy potential to the same scale for comparison purposes. In this plot we have projected the original points of
Lee and Secrest
1826 The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 TABLE V Energy Levels (in cm-') of CHz+ for the Carter and Handy Potential wave function
JK.IKI'
34(0 2' 0) 23(0 Oo 0) 19(0 4' 0) 33(0 3' 0) 27(0 1' 0) 14(0 5' 0) 31(0 4' 0) 30(0 22 0) lO(0 6' 0) 31(0 33 0) 27(0 53 0) 11(1 53 0)
om,
56(0 Oo 0) 1 l ( 0 40 0) lO(0 6O 0) 49(0 1' 0) 13(0 5' 0) 9(0 7l 0) 46(0 2' 0) 14(0 6' 0) 9(1 62 0) 7(0 S2 0) 44(0 33 0) 14(0 73 0) lO(1 7 3 0) 6(2 7' 0)
Om, 718.4 (718.34)
33(0 2O 0) 17(0 0' 0) 9(0 6O 0) 7(1 6O 0) 7(0 So 0) 28(0 3l 0) 7(1 7' 0) 19(0 1' 0) 6(0 9l 0) 8(0 7l 0) 6(1 9] 0) 27(0 42 0) 8(0 8' 0) 19(0 22 0) 6(2 S2 0) 8(1 S2 0) 5(1 10' 0) 28(0 53 0) 8(0 93 0) 18(0 3' 0) 6(2 93 0) 8( 1 93 0)
Om, 1611.8 (1611.04)
23(0 2O 0) 21 (0 40 0) 7(1 8 O O ) 6(0 So 0) 5(2 So 0) 5(1 loo 0) 25(0 3' 0) 6(2 9' 0) 17(0 5' 0) 5(0 9' 0) 7(1 9' 0) 25(0 4' 0) 16(0 6' 0) 7(1 lo2 0) 6(2 10' 0) 5(0 lo2 0)
Ow, 2771.8 (2770.67)
29(1 2' 0) 22(1 00 0) lO(1 40 0) 8(0 6O 0) 8(0 40 0) 25(1 1' 0) 23(1 3' 0) 13(0 5' 0) 25(1 2* 0) 17(1 42 0) 14(0 62 0) 24(1 33 0) 14(0 7' 0) 1 l(0 53 0)
Om, 3001.62 (2998.84)
0.0
energy
Ground Vibrational State (O,O,O) lo,, 14.7 (14.63)" 111, 118.6 (118.31) 110, 119.5 (119.05)
202, 44.1
303. 88.1
212, 147.3 211, 150.0
313, 190.1 312, 195.8
221, 361.3 220, 361.3
322, 406.0 321, 406.0 331, 692.9 330, 692.9
First Excited Vibrational State (0,1,0) 101, 732.9 (732.8) 111, 1056.5 (1056.9) 110, 1057.4 (1057.3)
202, 761.8
303, 805.2
212, 1085.0 2,,, 1087.7
313, 1127.5 312, 1133.1
221, 1465.4 220, 1465.5
322, 1509.9 321, 1509.9 331, 1932.7 330, 1932.5
Second Excited Vibrational State (0,2,0) lo,, 1626.4 (1625.1)
202, 1655.7
303, 1699.6
111, 2141.4 (2140.7) 110, 2142.0 (2140.9)
212, 2169.8 211, 2172.7
3,,, 2211.4 312, 2218.6
221, 2694.2 220, 2694.4
322, 2738.9 321, 2738.5 331, 3279.2 330, 3279.6
Third Excited Vibrational State (0,3,0) 1 2786.5 (2787.2)
202, 2816.1
303, 2860.2
3414.1 (3415.3) l l 0 , 3415.6 (3415.3)
212,3444.9 211, 3445.5
313, 3483.5 312, 3491.7
221, 4067.6 220, 4068.9
322, 41 13.5 3,,, 41 11.8
202, 3044.7
303, 3087.7
212, 3149.0 211, 3151.4
313, 3190.6 312, 3196.1
2,,, 3360.1 220, 3360.2
322, 3403.9 321, 3403.8
111,
Fourth Excited Vibrational State (1,0,0) l o ] ,3016.0 (3015.3)
8(0 7l 0) 6(2 3' 0)
8(2 4' 0) 8(2 4' 0) 11(1 53 0) 9(2 5' 0) 5(2 33 0)
I l l , 3120.9 l l o , 3121.6
331, 3685.2 330,3685.1
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1827
Structure of a Quasi-Linear Molecule CH2+ TABLE V (Continued)
wave function 36(0 2’ 1) 23(0 4’ 1 ) 22(0 0 0 1) 36(0 3l 1 ) 27(0 1’ 1) 18(0 5l 1 ) 35(0 42 1) 30(0 2’ 1) 13(0 62 1 ) 33(0 33 1) 32(0 53 1 ) lO(0 73 1 )
JK-IKI,
energy
Fifth Excited Vibrational State (0,0,1) Om, 3271.2 (3270.60) 101, 3285.7 (3287.1) 111, 110,
3387.8 (3387.6) 3388.7 (3390.0)
202, 3314.7
303,
212, 3416.1 211, 3418.8
3 , , , 3458.6
221, 3625.1 220, 3625.1
322, 3669.0 321, 3668.7
312,
3357.9
3464.0
330,
3949.1 3949.1
331,
Sixth Excited Vibrational State (l,l,O) 51(1 Oo 0) 9(0 6O 0) 7(0 8 O 0) 5(1 4O 0) 47(1 l 1 0) 12(0 71 0) 8(0 9l 0) 5(2 5I 0) 45(1 22 0) 13(0 S2 0) 7(0 lo2 0) 5(2 62 0)
Om, 3680.5 (3678.65)
l o , , 3694.6 (3691.0)
202, 3727.5
303,
3765.1
4020.8 (4021.0) 4020.8 (4021.7)
212, 4048.9 211,4050.4
313, 3 12,
4089.5 4094.7
221, 4424.8 220, 4425.1
322,
111, 110,
4468.6 321, 4468.2
“The numbers in parenthese are the Tennyson and Sutcliffe results of ref 4. Bartholomae, Martin, and Sutcliffe onto the plot. In parts c and d of Figure 1 we plot the SPF and Carter and Handy potential to higher energy. We see that the SPF potential predicts a shortening of the H-H distance as the molecule bends, while the Carter and Handy potential behaves somewhat strangely. This is of course in a region far from the original data points used in the fit. We point this out only to indicate that while the SPF gives an excellent fit to the data points it also behaves reasonably in the far field. Another difference of our SPF potential from the Carter and Handy potential to be noted is that the SPF barrier to linearity is higher (- 1142 cm-I) than the value predicted by Carter and Handy potential (-931 cm-I). We have investigated a large number of plots comparing the Carter and Handy and SPF surfaces both in the region of the a priori data point and in regions far from the original data. Near the a priori points the two fits look similar on most plots and far from the data they behave differently but both behave reasonably. However in Figure 2, a and b, we see another difference between the S P F and the Carter and Handy fit. Here we have plotted the asymmetric stretch against the symmetric stretch for the molecule held in the bent configuration. We have plotted the projection of the original data points onto these plots. The differences in the potential in the region of the original points are real. The S P F coordinates are not a convenient system in which to solve the rotation-vibration problem. It is convenient to transform the potential into the form of a polynomial in the coordinates t s , ta,and 7)b. In this form, the integrals of the potential matrix may be calculated for each mode independently. Reexpansion of the S P F potential in terms of the internal coordinates qs, qa,and qb is a nontrivial problem. A power series expansion around the minimum in these coordinates, or any other coordinates for that matter, has a small radius of convergence. A finite polynomial can fit the known potential points very nicely but the potential so obtained behaves poorly away from the well region in which the a priori potential is known. At large distances, the highest power terms carried in the calculation dominate and it is precisely these which are most poorly determined by the potential in the well region. Often these high-degree terms produce spurious deep wells separated from the a priori well by a high barrier. The high barrier is often not sufficient to prevent wave function density from building up in the spurious wells. We had encountered this problem in studies of other interaction potentials* and overcame them by using, rather than the internal coordinates themselves,
a function of the internal coordinates which behaves like a coordinate for small values and becomes constant at large values. We chose for purposes of expanding the potential the coordinates 6, =
tanh (aaaa) aa
6b =
tanh
(abqb)
(17)
(18)
ab
1 - e-w8
6, =
as
(19)
We could then develop the potential as a polynomial in these local functions. A rather high order polynomial in these functions is required to represent the potential accurately. Refitting the a priori potential in these functions would lead to too low an order of polynomial to be practical with the small number of points available on the potential surface. The SPF functions are more suited to the form of the potential and a highly acceptable SPF fit is possible. Thus we take the SPF fit to be a reasonable representation of the potential. Using the SPF potential, we may easily generate any number of points of the potential surface we please and fit this to as high an order as we please. We experimented with techniques for distributing the points to obtain reasonable fits. We found more points are needed to represent the potential surface correctly in the lower energy region compared to the higher energy region for this molecule. For this purpose, the potential points were distributed with constant intervals up to some cutoff energy in the vibrational coordinates. The higher energy points were distributed in a random manner by using a random number generator. Less points were generated at progressively higher energies. In Table 111, we give the error in this fit to the original data of Bartholomae, Martin, and Sutcliffe.6 While it shows the clear decrease of accuracy compared to the original SPF fit (Table 11), the lower energy points are still much better fit by this potential than by the potential of Carter and Handy. The potential derived through this procedure was shown to be superior to the potential obtained by direct fitting of the original Bartholomae, Martin, and Sutcliffe data points in the local vibrational coordinates in accuracy and stability. This procedure of transforming a potential
1828 The Journal of Physical Chemistry, Vol. 92, No. 7 , 1988
Lee and Secrest
TABLE VI: Energy Levels of CH2+for the SPF Potential wave function
JK&.
energy
Ground Vibrational State (O,O,O) 34(0 2O 0) 24(0 40 oj 18(0 0 0 0) 35(0 3' 0) 24(0 1' 0) 18(0 5' 0) 34(0 42 0) 27(0 22 0) 14(0 62 0) 31(0 0) 30(0 33 0) lO(0 73 0)
om, 0.0
101,
14.7
l,,, 116.7 ll0, 117.5
2,,, 44.0
303, 88.0
2,,, 145.3 2,,, 147.9
313, 188.2 312, 193.3
221, 370.8 220, 370.8
322, 415.3 321, 415.3 331, 721.6 330, 721.6
First Excited Vibrational State (0,1,0) 101, 853.4
51(0 Oo 0) 13(0 6O 0) 8(0 40 0) 44(0 1' 0) 13(0 7' 0) 11(0 5' 0) 7(1 7' 0) 41(0 22 0) 13(0 62 0) 11(0 82 0) 8(1 6I 0) 40(0 33 0) 13(0 73 0) 9(1 73 0) 9(0 93 0)
Om, 838.8
24(0 Oo 0) 23(0 2O 0) 9(0 8O 0) 7(1 8O 0) 6(0 4O 0) 5(1 6 O O ) 23(0 1' 0) 7(1 9' 0) 20(0 3, 0) 7(0 5' 0) 8(0 9l 0) 6(1 7l 0) 21(0 22 0) 7(0 lo2 0) 20(0 42 0) 6( 1 8* 0) 7(1 lo2 0) 4(2 S2 0) 7(0 62 0) 20(0 53 0) 6 ( i 113 0) 20(0 33 0) 5(2 93 0) 7(0 73 0) 5(0 11' 0) 6(1 9' 0)
Om, 1738.5
22(0 20 0) 14(0 4O 0) 13(0 6O 0) 6(1 10°O) 5(0 loo 0) 26(0 3' 0) 6(0 1' 0) 1 l ( 0 7' 0) 5(0 11' 0) 8(0 5' 0) 5(2 9' 0) 6(1 11' 0) 25(0 42 0) 6(0 22 0) 12(0 S2 0) 5(1 122 0) 7(0 6* 0) 5(2 lo2 0)
Om, 2864.7
33(1 2'0) 17(1 4OO) 16(1 0'0) 5(2 4O 0) 29(1 3' 0) 24(1 1' 0) 8(0 5' 0) 8(1 51 0) 7(0 7' 0) 27(1 22 0) 24(1 42 0) 9(0 62 0) 8(2 0) 7(0 82 0)
Om, 2938.0
I , , , 1140.1 l l 0 , 1141.1
202, 882.4
303, 926.0
2,,, 1168.6 2,,, 1171.5
313, 1211.3 312, 1217.1
221, 1531.6 220, 1531.6
322, 1576.1 321. 1576.1 331, 1988.5 330, 1988.5
Second Excited Vibrational State (0,2,0) 101, 1753.2
ill, 2243.0 1'0, 2244.2
202, 1782.5
303, 1826.6
212, 2271.3 2", 2275.0
313, 2313.9 312, 2321.2
221, 2767.0 220, 2767.0
322, 2812.3 321, 2812.1 331, 3321.6 330, 3321.3
Third Excited Vibrational State (0,3,0) lo,, 2879.7
202, 2909.5
l , , , 3478.7 l l 0 , 3479.9
212, 3507.0 2,'. 3511.3
Fourth Excited Vibrational State (1,0,0) lo,, 2952.5
111, 3053.5 110, 3054.3
303, 2954.6
221, 4091.5 220, 4091.4
322, 4140.4 321, 4140.0
202r2981.5
303, 3024.9
212, 3081.8 211, 3084.2
3,,, 3124.1 3,,, 3129.0
221,3310.4 2,,, 3310.4
322, 3354.3 32,. 3354.3
Structure of a Quasi-Linear Molecule CH2+
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1829
TABLE VI (Continued)
wave function
JK.,KI'
energy
28(1 33 0) 19(1 53 0) lO(0 73 0) 9(2 5j 0) 6(0 9) 0) 34(0 2' 1) 27(0 4O 1 ) 16(0 Oo 1) 36(0 3' 1) 22(0 1' 1 ) 21(0 51 1 ) 36(0 42 1 ) 26(0 22 1) 17(0 62 1) 34(0 53 1) 29(0 33 1) 13(0 73 1)
331, 3663.2 330, 3663.2
Fifth Excited Vibrational State (0,O.l) Om, 3226.4 101, 3240.8 111, 3335.9 1 io, 3336.7
202, 3269.7
303, 3312.9
212, 3364.0 2 , , , 3366.6
313, 3406.3 312, 341 1.3
2,,, 3579.0 2,,, 3579.0
322. 3622.5 321, 3622.5 331, 3916.8 330, 3916.8
Sixth Excited Vibrational State (l,l,O) 48(1 Oo 0) 8(0 8 O 0) 6(1 4' 0) 5(0 6O 0) 5(2 6O 0) 41(1 1 ' 0 ) 7(1 5, 0) 9(0 9, 0) 6(2 5l 0)
000,
3756.8
function from arbitrary coordinates to local vibrational coordinates was repeated for the Carter and Handy potential to check the reliability of this method in predicting the rotation-vibration states of this molecule. The fit to this surface is also included in Table 111. Calculation and Results
We have carried out calculations for J = 0, 1 , 2 , and 3. The basis functions have been divided according to the parity of the sum of asymmetric stretch and bend quantum number, to simplify calculation, as the even set does not couple with the odd set. In Table IV, we show the number of basis functions used in each case. The singlet and triplet designations refer to the nuclear spin multiplicities of the H atoms. The basis set was converged to the point that adding a further function lowered the energy of the lowest five states by less than 0.1 cm-I and the next five states by less than 0.5 cm-I. The procedure of calculating the Hamiltonian matrix elements and the diagonalization scheme were explained in detail in ref 8. In Table IV we also give the exponential parameters used in the calculation. These exponential parameters were optimized to give the lowest energy with a small basis set. When the converged basis set is used, the energy levels are insensitive to the values of exponents. Optimizing the parameters leads to rapid convergence of the wave-function expansion. In Table V, we have tabulated the energy levels for the Carter and Handy potential which was transformed to our local functions by the technique described above. The energy is given in wavenumbers above the zero-point energy. Also, in Table V, we present the energy levels of CH2+calculated by Tennyson and Sutcliffe4 using the Carter and Handy potential. For the states reported by Tennyson and Sutcliffe, our results compare very well with their results, confirming that our procedure of transforming the potential to local vibrational coordinates is accurate. In Table VI, we present the results for the SPF potential. In both Tables V and VI the rotational states are labeled in the J K _ , Knotation , of Townes and S c h a ~ l o w . ' ~The even states are those with K-, + n, even and the singlet states are those with K-I K , n, even, the odd and triplet states being given by the odd value of the corresponding quantity. The energy levels are grouped by vibrational level with all of the rotational levels of each
+ +
(17) Townes, C. H.; Schawlow, A. L. Microwave Spectroscopy; McGraw-Hill: New York, 1955.
3771.2
202, 3799.8
303, 3842.9
l I 1 ,4063.4 4064.4
212, 4091.2 2 , , , 4094.4
313, 4133.0 312, 4139.4
101,
110,
vibrational state grouped together. The higher rotational states of some levels overlap, as can be seen from Tables V and VI. The column labeled wave function gives the percent composition of the wave function. This number is the square of the dominant coefficients of the normalized wave function multiplied by 100 and it is followed by the vibrational quantum numbers of the basis functions. These compositions are essentially independent of rotational state J . We have included all basis functions which contribute 5% or more. It is clear that the molecule behaves like a prolate molecule. When K # 0 the levels group as doublets with identical coefficients for the linear combinations. Of course, in a doublet pair, the basis functions are different. For example, the 221and 2,, state basis function (0 42 0) is the symmetric linear combination of the K = 2 and K = -2 functions for 220and the antisymmetric for 221. The first excited rotational state doulet of each vibrational state (K-l = 1) is well split and all higher states K-, > 1 are degenerate to the accuracy of the calculation. There are a few cases in the table where the states appear split, but these higher energies are only converged to cm-I. To check this, a few calculations were repeated with larger basis sets and the levels became equal to 0.1 cm-I. The major contribution to the ground vibrational state comes from an excited bend basis function. This is to be expected since the equilibrium configuration of the molecule is bent. The first three vibrationally excited states are bend states for both the SPF and Carter and Handy potentials. The fourth excited vibrational state is a symmetric stretch and the fifth excited vibrational state is an asymmetric stretch. The behavior of rotational states for each vibrational level is interesting. We notice in the ground vibrational state, the separation between K states in each vibrational level are nearly independent of rotation level. For example, 2 , , - 202 = 312- 303 = 104. In the excited bend states, these differences get much larger. 2,, - 2,, = 3,, - 303= 290 for the first excited bend. And for the second excited bend, 2 , , - 202 = 312- 303 = 490. As we go to higher excited bending states the molecule becomes more linearlike. The fourth excited state is a symmetric stretch and for this state 2,' - 2,, = 312- 303= 105, just like the ground state. The fifth excited state is an asymmetric stretch and 2 , - 203 = 3 ) , - 303 = 99, a bit less than the ground state. The sixth state seems to be a combination symmetric stretch and bend and has 2!, ; 2,, = 312- 3,, = 290, characteristic of a single bend excitation. The results for the Carter and Handy potential are
J . Phys. Chem. 1988, 92, 1830-1835
1830
TABLE VII: Comparison of Band Origins and Zero-Point Energy (in cm-') for the CH,' Molecule this work HC SPF TS" HCh BMS' 2935 2999 3001.62 2938.02 2998.8 VI 908 718.3 718 718.43 838.81 "2 3162 3271 3271.20 3226.40 3270.7 "3 zero-point energy 3662.12 3552.69 3660.18 3662.1 3541.2 ~
~
a
~
~
Reference 4. bReference 3. Reference 6
similar. This state was assigned by Tennyson and Sutcliffe to the 4th excited bend state (0 4 0). It is difficult to assign this state from the basis function composition alone, and the higher rotational state levels give a clear indication of the vibrational state. The CH2+is of C,, symmetry and will be separated into two nuclear spin states depending on whether the H nuclear spins form a triplet or singlet state. The singlet state occurs when K , + K-, n, is even, and the triplet state occurs for odd values of this quantity. Spectroscopic transitions between singlet and triplet states are rare. The only strong dipole selection rule requires transitions between states with K-, n, differing in parity. Other than these symmetry-imposed selection rules, any transition appears to be allowed. In Table VII, we compare our results for the band origins and zero-point energy for this molecule to previous calculation^.^^^^^ While our results using the Carter and Handy fit to the potential agrees well with other calculations on the same potential energy surface, the results for the SPF-type potential differs substantially from those made on the Carter and Handy surface. The S P F surface fits the a priori potential points better than the Carter and Handy surface, and we see that the results are somewhat more in agreement with the Bartholomae, Martin, and Sutcliffe calculation which appeared in their original paper on this system. The Carter and Handy surface is of the type developed by Sorbie and Murrell16 for general triatomics. It is interesting to note that the SPF-type potential was found to predict the vibrational energies of the molecules more accurately than the Sorbie-Murrell-type potential, especially for the molecules with large amplitude of the vibrational motion, also in calculations by Carney, Curtiss, and Langhoff.14
+
+
Conclusion
We have shown that using our linear molecule Hamiltonian it is indeed possible to obtain a rotation-vibration spectra for a floppy nonlinear molecule. It is difficult to identify the vibrational states for molecules like this with such mixed wave functions in the harmonic oscillator basis set. However, with enough rotational states one can assign the vibrational states properly and we were able to identify the (1 1 0) state through properties of its rotational spectra. This assignment is not at all obvious from an analysis of the basis function expansion alone. This is very likely due to the fact that our basis functions are harmonic oscillator functions and the molecule is far from harmonic. The S P F method gave a dramatic improvement in the fit of the potential to the original a priori potential points for the molecule. The difference in the potential gave a noticeable difference in the rotation-vibration energy level structure. This difference is particularly noticeable in the bending mode. The S P F fit gives a higher frequency bend and a larger barrier to linearity than the Carter and Handy fit. The results for the SPF fit are in much better agreement with the original band origins of Bartholomae, Martin, and Sutcliffe than those obtained by using the Carter and Handy fit to the data. We have also presented a method of transforming a potential function from an arbitrary coordinate system to the local coordinates of eq 17-19. We have applied this technique to both the Carter and Handy potential and the S P F potential, and it is generally applicable for any representation of the potential. We have neglected the Renner-Teller effect as was done in the earlier calculations. It is of course necessary to correct for this effect to obtain results to compare directly with experiment, but since in the present case we are testing a fit to ab initio points this effect has no relevance to the present results.
Acknowledgment. This work was supported by a grant from the National Science Foundation with computer time from NCSA at the University of Illinois. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for a grant. Registry No. CH2+,15091-72-2.
Photodecomposition of Europium( I I I ) Acetate and Formate in Aqueous Solutions Akira Matsumoto* and Nagao Azuma Laboratory of Chemistry, Faculty of General Education, Ehime University, Bunkyo-cho, Matsuyama 790, Japan (Received: April I , 1986; In Final Form: September 4, 1987) The products obtained in the photodecompositionof europium(II1) acetate and formate were determined quantitatively under various pH conditions. The two photoprocessesand the subsequent radical reactions have been found to proceed stoichiometrically with a cyclic redox reaction of the Eu3+/Eu2+couple being the key process. In the acetate system, (1) redox reactions of methyl radical with Eu2+and Eu" aquo ions are supported to occur with formation of methane and methanol, respectively; ( 2 ) carboxymethyl radical produced through hydrogen abstraction from acetic acid by the methyl radical probably regenerates acetic acid via the intermediate formed through the radical capture reaction by Eu2+ ion, followed by acidolysis reaction. In the formate system, (1) it was confirmed that the photoprocess is followed by a chain reaction in which carboxyl radicals generated in the primary process and hydrogen atoms produced through reduction by illuminated Eu2' ion behave as chain carriers; (2) evolution of carbon dioxide gas is mostly ascribed to the disproportionationof carboxyl radical and the reduction of Eu3+ion by carboxyl radical. In both systems, minor products are formed by redox and disproportionation radical reactions.
Introduction
R-CO2H
The primary process of the photochemical reaction of complexes of one-electron-oxidizing metal ions with aliphatic carboxylic acids is known to be a charge transfer from ligand to metal (CTTM), which gives rise to the products shown by eq 1' and 22 (1) (a) Sheldon, R. A.; Kochi, J. K. J . Am. Chem. SOC.1968, 90, 6688. (b) Kochi, J. K.; Sheldon, R. A.; Lande, S S Tefrahedron 1969, 25, 1197.
0022-3654/88/2092-1830$01.50/0
I
+
hv
-
Mn+
H-C02H
I
+
hv
'R
-
+
COn
'C02H
+
+ H+ + H+
+
Mn-'
Mn-'
(1)
(2)
Mn+
Equation 2 is usually followed by the redox reaction of the strongly reducible carboxyl radical, so that the overall net reaction
0 1988 American Chemical Society
