18088
J. Phys. Chem. 1996, 100, 18088-18092
High-Level ab Initio Calculation of the Rotation-Vibration Energies in the c˜ 1A1 State of Methylene, CH2 P. R. Bunker* Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada
Per Jensen FB9-Theoretische Chemie, Bergische UniVersita¨ t-Gesamthochschule Wuppertal, D-42097 Wuppertal, Germany
Yukio Yamaguchi and Henry F. Schaefer III Center for Computational Quantum Chemistry, UniVersity of Georgia, Athens, Georgia 30602 ReceiVed: July 8, 1996; In Final Form: August 16, 1996X
For the third excited electronic state (c˜ 1A1) of the methylene radical, CH2, we calculate the electronic potential energy surface using a high-level ab initio method and the rotation-vibration energies using a variational technique with a large rotation-vibration basis set. The potential energy surface is calculated at a carefully selected grid of 48 nuclear geometries that cover all types of combination of stretching and bending deformations to energies more than 20 000 cm-1 above that of the equilibrium configuration. We fit an analytical function, in which we vary 23 parameters, through the points and find that the state is almost linear with an equilibrium angle of 172.7° and a barrier to linearity of only 6 cm-1. The potential energy points were determined by employing the complete active space self-consistent-field (CASSCF) reference second-order configuration interaction (SOCI) method. The CASSCF and SOCI wave functions were constructed following the second eigenvector of the corresponding CI Hamiltonian matrices. It is well-known that theoretical treatments of higher lying states in the same symmetry are substantially tedious and complicated. The basis set used [TZ3P(2f,2d)+2diff] was triple-ζ plus triple polarization with two sets of higher angular momentum functions and two sets of diffuse functions. We have used the variational MORBID procedure to calculate the rotationvibration energies. Because of the peculiar shape of the bending part of the potential surface, some very large bending force constants f(i) 0 are obtained, and this has necessitated the use of very large basis sets in the MORBID calculation in order to achieve acceptable convergence.
I. Introduction In ref 1 a series of ab initio calculations of the equilibrium geometries, harmonic vibrational frequencies, and dipole moments of the X ˜ 3B1, a˜ 1A1, b˜ 1B1, and c˜ 1A1 electronic states of the methylene radical, CH2, were reported. The c˜ state was found to be nearly linear with an equilibrium bond angle of 171.6° at the highest level of ab initio calculation performed with geometry optimization (TZ3P(2f,2d)+2diff TCSCF-CISD). The equilibrium geometry was obtained by using an analytic gradient technique. In a subsequent paper2 many points on the c˜ state surface were calculated at the same level of ab initio theory and an analytical potential energy function was fitted through the points; in this fitting the equilibrium geometry was constrained at that determined by the ab initio gradient technique. With the analytical potential energy function the rotation-vibration energies were calculated using the MORBID Hamiltonian and computer program.3-5 The aim was to see if we could thereby assign as being the c˜ 1A1 r a˜ 1A1 electronic band system the unidentified bands reported between 27 000 and 30 000 cm-1 by Herzberg and Johns6 in their paper on CH2. We showed that our results did not support this assignment. However, the c˜ 1A1 electronic state of CH2 is not the lowest state of this symmetry for methylene, and the accuracy of ab initio methods for such excited state calculations must be tested. X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01993-4 CCC: $12.00
The c˜ state of the CH2 provides a convenient example for making such a test, and with this in mind the present paper uses a higher level of ab initio theory to determine the energies at 48 points on the potential energy surface. The potential energy points were determined by employing the complete active space self-consistent-field (CASSCF) reference secondorder configuration interaction (SOCI) method. The CASSCF and SOCI wave functions were constructed following the second eigenvector of the corresponding CI Hamiltonian matrices. It is well-known that theoretical treatments of higher lying states in the same symmetry are substantially tedious and complicated. The basis set used [TZ3P(2f,2d)+2diff] was triple-ζ plus triple polarization with two sets of higher angular momentum functions and two sets of diffuse functions. We fit the same analytical expression for the potential energy function through the points as used in ref 2 and again use the MORBID Hamiltonian and computer program to calculate the rotationvibration energies. No analytical derivative method has been implemented at the level of ab initio calculation performed here, and so the equilibrium structure and energy are determined from the analytical function. As in ref 2 we find that the bending part of the potential surface has an unusual shape, requiring many large parameters in our analytical function. This causes problems both in fitting the function through the points and in obtaining converged rotation-vibration energies. These problems are discussed here. © 1996 American Chemical Society
c˜ 1A1 State of Methylene
J. Phys. Chem., Vol. 100, No. 46, 1996 18089
II. The ab Initio Calculation The basis set used in this study was triple-ζ plus triple polarization with two sets of higher angular momentum functions and two sets of diffuse functions [TZ3P(2f,2d)+2diff]. The basis sets of triple-ζ (TZ) quality were derived from Dunning’s triple-ζ contraction7 of Huzinaga’s primitive Gaussian functions,8 and they are designated (10s6p/5s3p) for C and (5s/3s) for H. The orbital exponents of the polarization functions were Rd(C) ) 3.00, 0.75, 0.1875 and Rp(H) ) 3.00, 0.75, 0.1875; those of the higher angular momentum functions were af(C) ) 1.60, 0.400 and Rd(H) ) 2.00, 0.50, and those of the diffuse functions were Rp(C) ) 0.033 89, 0.012 53 and Rs(C) ) 0.048 12, 0.016 69, Rs(H) ) 0.030 16, 0.009 247. Six Cartesian d-like and 10 Cartesian f-like functions were used throughout. This basis set comprised 112 contracted Gaussian functions with a contraction scheme of C (12s8p3d2f/7s5p3d2f) and H (7s3p2d/5s3p2d). The zeroth-order description of the c˜ 1A1 state was obtained as the second eigenvector of the complete active space selfconsistent-field (CASSCF)9-11 secular equation. The CAS space included the six valence electrons in the six valence molecular orbitals (MOs) and consisted of 56 configuration state functions (CSFs) in C2V symmetry (for bent symmetric stretching and bending modes), 60 CSFs in C2V symmetry (for linear asymmetric stretching modes), and 100 CSFs in Cs symmetry (for bent asymmetric modes). The second-order configuration interaction (SOCI) technique,12 which is single and double excitation configuration interaction relative to the CASSCF references, was employed to include correlation effects in a larger CI space. The CASSCF-SOCI energy for the c˜ 1A1 state was determined again following the second root of the SOCI Hamiltonian matrix. In all the SOCI procedures one core (C 1s-like) orbital was frozen and the corresponding virtual (C 1s*like) orbital was deleted. The numbers of CSFs for the SOCI wave functions were 296 371 in C2V symmetry (for bent symmetric stretching and bending modes), 302 622 in C2V symmetry (for linear asymmetric stretching modes), and 591 780 in Cs symmetry (for bent asymmetric modes), respectively. All computations were performed using the PSI 2.0 suite of ab initio quantum mechanical programs.13 III. The Calculation of the Rotation-Vibration Energies To calculate the rotation-vibration energies, we need to interpolate and extrapolate the internuclear potential energy function V to nuclear geometries at other than the 48 electronic energies obtained here. To do this, using the MORBID approach,3-5 we write V as the following analytical function of the two bond lengths and the bond angle:
r32
R
Eb
∆Ec
1.1047 1.0640 1.0640 1.0640 1.0640 1.0640 1.0640 1.0640 1.0640 1.3640 1.3640 1.3640 0.8640 0.8640 0.8640 1.3140 1.3140 1.3140 0.8890 0.8890 0.8890 1.2240 1.2240 1.2240 0.9390 0.9390 0.9390
1.1047 1.0640 1.0640 1.0640 1.0640 1.0640 1.0640 1.0640 1.0640 1.3640 1.3640 1.3640 0.8640 0.8640 0.8640 1.3140 1.3140 1.3140 0.8890 0.8890 0.8890 1.2240 1.2240 1.2240 0.9390 0.9390 0.9390
102.2984 180.0000 160.0000 140.0000 130.0000 120.0000 110.0000 100.0000 80.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000
-38.901 451 54 -38.985 545 60 -38.984 550 49 -38.971 405 77 -38.958 289 09 -38.940 960 01 -38.919 353 71 -38.893 218 65 -38.825 941 95 -38.913 078 38 -38.916 313 60 -38.884 969 72 -38.897 240 16 -38.894 289 49 -38.842 079 15 -38.931 001 06 -38.933 159 22 -38.900 072 53 -38.921 261 35 -38.918 514 14 -38.867 380 60 -38.959 787 84 -38.960 528 26 -38.923 882 96 -38.995 649 25 -38.953 605 01 -38.904 396 08
0.0 12.0 -4.9 8.3 -12.2 -1.0 -3.8 0.8 0.0 4.2 -8.9 -4.5 6.2 -9.7 0.5 -0.7 12.2 6.8 -8.1 18.8 -2.0 -3.8 -6.8 0.7 -6.1 -8.5 -1.3
1.3640 1.3640 1.3640 1.3140 1.3140 1.3140 1.2640 1.2640 1.2640 1.0640 1.0640 1.0640 1.0640 1.0640 1.0640 1.1500 1.1500 1.1500 1.0000 1.0000 1.0000
0.8640 0.8640 0.8640 0.8890 0.8890 0.8890 0.9640 0.9640 0.9640 0.9250 0.9250 0.9250 1.2640 1.2640 1.2640 1.2500 1.2500 1.2500 0.9500 0.9500 0.9500
180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000 180.0000 160.0000 120.0000
-38.900 286 80 -38.900 436 55 -38.862 794 65 -38.922 649 50 -38.922 456 70 -38.883 278 61 -38.955 742 81 -38.955 438 45 -38.915 446 10 -38.966 237 05 -38.964 662 37 -38.918 264 63 -38.965 871 55 -38.965 992 71 -38.927 404 30 -38.964 530 00 -38.964 995 88 -38.927 492 68 -38.969 669 04 -38.967 916 74 -38.920 367 37
7.4 0.5 -5.6 -7.4 -2.8 4.6 -11.7 8.2 4.6 7.2 -7.5 6.8 -0.3 8.5 -0.9 -0.5 -4.2 -1.7 3.4 1.5 2.1
a Bond lengths r and r in Å and bond angles R in deg. b Ab initio 12 32 energies in hartrees (1 hartree corresponds to 219 474.630 6 cm-1). c Residuals (ab initio - fitted) in cm-1.
N
jek
∑ Fjkm(Fj)yjykym + jekemen ∑ Fjkmn(Fj)yjykymyn jekem
r12
and
V(∆r1,∆r3,Fj) ) V0(Fj) + ∑ Fj(Fj)yj + ∑ Fjk(Fj)yjyk + j
TABLE 1: Ab Initio Energies for the c˜ 1A1 State of CH2 and the Residuals (ab Initio - Fitted) Obtained after Fitting the Function of Eqs 1-4 through the Pointsa
(1)
where j, k, m, and n can be 1 or 3, yj ) 1 - exp(-aj∆rj), the aj are molecular constants, ∆rj ) rj - rje, and rje is the equilibrium value of rj, the distance between the “outer” nucleus j and the “center” nucleus 2. The quantity Fj is the instantaneous value of the bond angle supplement (see Figure 1 of ref 3). The Fjkm... expansion coefficients of eq 1 are functions of Fj defined as
(0) (i) Fjk...(Fj) ) fjk... + ∑ fjk... (cos Fe - cos Fj)i
(3)
i)1
where Fe is the equilibrium value of Fj and the f(i) jk... are further expansion coefficients. The function Fjk(Fj) has N ) 3, Fjkl(Fj) has N ) 2, and Fjklm(Fj) has N ) 1. The function V0(Fj) is the potential energy for the molecule bending with bond lengths fixed at their equilibrium values, and we parametrize it as 8
j )i V0(Fj) ) Ve + ∑ f(i) 0 (cos Fe - cos F
(4)
i)2
4
Fj(Fj) ) ∑ fj(i)(cos Fe - cos Fj)i i)1
(2)
where the f(i) 0 are expansion coefficients and Ve is the energy at equilibrium. For a symmetrical triatomic molecule, relations,
18090 J. Phys. Chem., Vol. 100, No. 46, 1996
Bunker et al.
TABLE 2: Parameters Obtained by Fitting the Potential Function of Eqs 1-4 through the ab Initio Points Given in Table 1a parameter
value
parameter
value
Ve/hartree a1 Re
-38.985 1.6c 172.70(40) 1.067 616(87) 96 905(1737) -217 641(9712) 295 153(21025) -203 296(19 796) 56 013(6716) -9910(433) 10 668(1414) -10 975(1227)
f(0) 11 f(1) 11 f(2) 11 f(0) 13 f(2) 13 f(0) 111 f(1) 111 f(2) 111 f(0) 113 f(1) 113 f(0) 1111 f(0) 1113
61 829(79) -15 682(680) 16 802(1285) -3738(89) 12 935(158) -11 393(309) -43 916(3724) 77 855(7078) -2025(246) 3145(799) 2933(445) -1589(345)
re12 f(2) 0 f(3) 0 f(4) 0 f(5) 0 f(6) 0 f(1) 1 f(2) 1 f(3) 1
644(23)b
a Units are bond lengths in Å, bond angle in deg, and force constants in cm-1. b One standard error given in parentheses in units of the last figure quoted for the parameter. c Held fixed in the fitting.
(i) such as f(i) 11 ) f33, exist so that V is symmetric with respect to interchange of ∆r1 and ∆r3. This expression for V has been developed in a way that involves the yi coordinates since the stretching basis functions used in the MORBID approach are Morse oscillator functions; using such basis functions, analytic expressions for the matrix elements of yi and its powers exist. The adjustable parameters (i) in V that give the function its flexibility are the f(i) 0 , fj , and (i) e e fjk.... These parameters, together with Ve, Fe, r1 ) r3 , and a1 ) a3, constitute 35 adjustable parameters for a symmetrical triatomic molecule. They are determined by the least squares fitting of V to the 48 values determined by ab initio calculation. In the fitting for the c˜ state of CH2 to the 48 ab initio points we have not been able to constrain the equilibrium energy and the equilibrium bond lengths and bond angle to the values obtained directly in the ab initio calculation since standard analytic derivative methods could not be used in the ab initio calculation. The equilibrium geometry is thus the structure at the minimum of the determined analytical function. We have constrained a1 ) a3 ) 1.6 Å-1 based on previous experience.14 The 48 points include a point (the third one in Table 1) at the equilibrium geometry of the a˜ state, and this equilibrium geometry comes from ref 1. By varying 23 parameters, we achieve a fitting with a standard deviation of 9 cm-1. The variation of further parameters does not achieve a statistically meaningful improvement. The values of the fitted parameters are given in Table 2, and the residuals (ab initio - fitted) are included in Table 1. This fitting reproduces the interesting result obtained in ref 2 of very large values for the bending force constants f(i) 0 , which is caused by the unusual shape of the bending potential. We have used the parameters of Table 2 as input data in order to calculate the rotation-vibration energies using the MORBID Hamiltonian and computer program.3-5 In Table 3 we give the rovibronic energies calculated for the J ) l ) 0 and J ) l ) 1 f-levels up to 15 000 cm-1 above the lowest level of the c˜ state. Because the pattern of subband origins of the c˜ 1A1 r a˜ 1A1 electronic band system will be those appropriate for a linearbent electronic transition, we use linear molecule notation in Table 3. To aid in recognizing the rotation-vibration energy level pattern for the lowest vibrational levels in the c˜ state should appropriate data become available, we give the rovibronic energies up to J ) 5 for the lowest six vibrational states using bent molecule notation in Table 4.
TABLE 3: Term Values (in cm-1) of J ) l ) 0 and J ) 1, l ) 1 f-Levels Below 15 000 cm-1 Using the W2 and l Quantum Numbers Appropriate for a Linear Molecule J ) 0, l ) 0 Γrv ) A1
Γrv ) B2
V1
Vl2
V3
E
0 0 1 0 1 0 2 0 1 2 0 0 1 3 1 2 0 0 3 1 1 4 2 2 0 3 0 1 0 4 1 2 2 0 5 3 0
00 20 00 40 20 60 00 00 40 20 80 20 60 00 00 40 40 100 20 20 80 00 60 00 60 40 120 40 00 20 100 20 80 80 00 60 20
0 0 0 0 0 0 0 2 0 0 0 2 0 0 2 0 2 0 0 2 0 0 0 2 2 0 0 2 4 0 0 2 0 2 0 0 4
(3923.2)a 1760.7 3058.6 3837.3 4764.9 6026.2 6051.8 6724.2 6805.1 7698.7 8265.6 8350.1 8963.5 8979.5 9559.0 9696.9 10 298.7 10 528.0 10 560.2 11 119.5 11 195.2 11 819.2 11 838.0 12 322.1 12 380.4 12 510.0 12 798.4 13 021.6 13 161.0 13 344.1 13 446.7 13 819.4 14 046.0 14 524.6 14 590.6 14 614.9 14 646.3
V1
Vl2
V3
E
0 0 1 0 1 2 0 1 0 2 0 0 3 1 2 1 0 3 0 1 1 4 2
00 20 00 40 20 00 60 40 00 20 80 20 00 60 40 00 40 20 100 20 80 00 60
1 1 1 1 1 1 1 1 3 1 1 3 1 1 1 3 3 1 1 3 1 1 1
3397.7 5090.7 6338.5 7103.9 7971.3 9205.6 9239.1 9937.4 9982.6 10 774.5 11 428.9 11 535.6 11 995.4 12 048.2 12 682.5 12 724.6 13 418.4 13 496.7 13 649.8 14 194.3 14 236.9 14 701.8 14 767.8
J ) 1, l ) 1f Γrv ) A1 V1
Vl2
0 0 1 0 1 2 0 0 1 2 0 0 3 1 1 2 0 3 0
11 31 11 51 31 11 71 11 51 31 31 91 11 71 11 51 51 31 111
a
V3 1 1 1 1 1 1 1 3 1 1 3 1 1 1 3 1 3 1 1
Γrv ) B2 E 4108.8 6028.5 7019.1 8111.3 8891.8 9851.5 10 267.1 10 593.1 10 942.3 11 671.8 12 384.8 12 456.3 12 601.6 13 080.9 13 292.6 13 685.6 14 351.8 14 364.9 14 659.9
V1
Vl2
V3
E
0 0 1 0 1 2 0 0 1 2 0 0 3 1 1 2 0 3 0 1 1 4 2 2 0 3 0 0 1 4 1 2
11 31 11 51 31 11 71 11 51 31 31 91 11 71 11 51 51 31 111 31 91 11 71 11 71 51 11 131 51 31 111 31
0 0 0 0 0 0 0 2 0 0 2 0 0 0 2 0 2 0 0 2 0 0 0 2 2 0 4 0 2 0 0 2
759.4 2742.6 3792.6 4883.4 5734.3 6755.6 7090.5 7387.7 7845.7 8651.3 9243.7 9326.2 9645.7 10 033.9 10 189.9 10 730.9 11 269.3 11 490.1 11 574.3 11 996.5 12 256.7 12 457.5 12 901.3 12 919.3 13 376.2 13 533.8 13 723.6 13 826.0 13 989.8 14 243.6 14 493.8 14 673.5
Zero-point energy in parentheses.
c˜ 1A1 State of Methylene
J. Phys. Chem., Vol. 100, No. 46, 1996 18091
TABLE 4: Rovibronic Term Values (in cm-1) for the c˜ 1A1 State of CH2 Using the Quantum Number Notation Appropriate for a Bent Moleculea (V1, Vbent 2 , V3)
J Ka Kc (0,0,0)
(0,1,0)
(1,0,0)
(0,0,1)
(0,2,0)
(1,1,0)
5 5 5 5 5 5 5 5 5 5 5
5 5 4 4 3 3 2 2 1 1 0
0 1 1 2 2 3 3 4 4 5 5
4661.374 4661.374 3677.670 3677.670 2723.115 2723.115 1811.062 1811.113 964.244 960.208 217.986
6969.502 6969.502 5920.565 5920.565 4921.842 4921.831 3927.748 3928.034 2948.982 2939.939 1979.321
7604.126 7604.126 6638.164 6638.164 5703.931 5703.930 4814.898 4814.948 3994.110 3990.054 3273.892
7868.506 7868.506 6901.765 6901.765 6013.424 6013.423 5126.844 5126.895 4312.658 4309.030 3612.462
9255.437 9255.436 8226.827 8226.838 7126.892 7126.703 6115.447 6117.259 5091.345 5075.279 4055.884
9885.037 9885.037 8900.167 8900.167 7870.238 7870.223 6896.188 6896.504 5937.055 5928.067 4980.990
4 4 4 4 4 4 4 4 4
4 4 3 3 2 2 1 1 0
0 1 1 2 2 3 3 4 4
3605.197 3605.197 2650.548 2650.547 1738.461 1738.482 890.971 888.277 145.357
5856.741 5856.741 4852.308 4852.305 3856.430 3856.554 2875.166 2869.132 1906.555
6567.078 6567.078 5632.686 5632.686 4743.558 4743.580 3922.037 3919.329 3202.401
6846.052 6846.052 5933.492 5933.492 5052.558 5052.586 4239.728 4237.305 3540.891
8145.417 8145.418 7066.600 7066.522 6048.438 6049.296 5016.931 5006.180 3983.119
8818.530 8818.530 7803.595 7803.591 6826.398 6826.536 5864.523 5858.526 4909.431
3 3 3 3 3 3 3
3 3 2 2 1 1 0
0 1 1 2 2 3 3
2592.465 2592.465 1680.346 1680.353 832.327 830.709 87.230
4796.674 4796.674 3799.305 3799.347 2816.088 2812.465 1848.325
5575.661 5575.661 4686.453 4686.460 3864.353 3862.727 3145.184
5868.046 5868.046 4993.044 4993.056 4181.357 4179.902 3483.610
7021.564 7021.541 5994.850 5995.162 4957.407 4950.936 3924.926
7750.414 7750.413 6770.492 6770.539 5806.473 5802.872 4852.167
2 2 2 2 2
2 2 1 1 0
0 1636.740 3756.418 4643.605 4948.353 5954.666 1 1636.741 3756.427 4643.606 4948.356 5954.733 1 788.329 2771.765 3821.075 4137.565 4912.772 2 787.520 2769.953 3820.261 4136.837 4909.528 2 43.621 1804.643 3102.258 3440.635 3881.294
6728.519 6728.528 5762.921 5761.119 4809.209
1 1 0 1 1 1 1 0 1 0 0 0
758.990 2742.210 3792.216 4108.363 4883.021 5733.879 758.721 2741.605 3791.945 4108.120 4881.938 5733.278 14.542 1775.517 3073.634 3411.979 3852.212 4780.565 0.0
1760.952 3059.320 3397.649 3837.673 4766.242
Linear molecule notation has V2 ) Vlinear ) 2Vbent + Ka, and l ) 2 2 Ka; the e-levels have (J - Kc) even, and the f-levels have (J - Kc) odd. a
Because of the large values of some of the parameters, convergence of the energies was difficult to achieve, and it was necessary to use rather large basis sets. In the final calculation for Table 3 we used a basis set (see ref 3) in which the stretching problem was prediagonalized with Morse oscillator functions |n1n3〉 having n1 + n3 e NStretch ) 16. In constructing the final rotation-vibration matrices we used the NBend ) 29 lowest bending basis functions, the NA ) 26 lowest stretching basis functions of A1 symmetry, and the NB ) 21 lowest stretching basis functions of B2 symmetry. The largest J ) 1 matrices that had to be diagonalized were 1363 × 1363. By comparison with the results of calculations using smaller basis sets, we determine that the highest levels in Table 3 are converged to better than 5 cm-1, and the lowest to better than 1 cm-1. IV. Discussion and Conclusions Using this same level of ab initio method throughout, we obtain Te(a˜) ) 3300 cm-1, Te(b˜ ) ) 11 664 cm-1, and Te(c˜) ) 20 815 cm-1 from ref 1. It should be noted that these energy separations were evaluated at the TZ3P(2f,2d)+2diff TCSCFCISD optimized geometries. Using the TCSCF-CISD method with the same basis set, the corresponding energy separations were Te(a˜) ) 3234 cm-1, Te(b˜ ) ) 11 620 cm-1, and Te(c˜) )
Figure 1. Bending potential V0(Fj) (eq 4, full curve) and the functions V(n)(Fj) (eq 5, dashed curves) for c˜ 1A1 CH2.
21 080 cm-1, respectively, as reported in refs 1 and 2. It is seen that the two sets of Te values at the two different levels of theory are consistent with each other. The largest deviation is only 265 (21 080 - 20 815) cm-1 for the Te(c˜) value. Bauschlicher and Yarkony15 used the TCSCF-CISD method with a double-ζ plus polarization (DZP) basis set and estimated the Te(c˜) value to be 23 400 cm-1 (2.9 eV), while Rice and Handy16 used the CASSCF method employing a DZP plus diffuse functions basis set and obtained the Te(c˜) value of 22 840 cm-1 (2.832 eV). The CASSCF-SOCI method employed in this research includes a significantly higher amount of correlation effects and predicts the 2600-2000 cm-1 lower Te(c˜) value than their values. The vertical energy of the b˜ 1B1 state above the a˜ 1A1 state at the a˜ state equilibrium geometry is determined to be 13 791 cm-1 (14 027 cm-1 in ref 2), in agreement with the region (11 000-19 000 cm-1) in which the b˜ r a˜ band system is observed in ref 6. However, the vertical energy of the c˜ 1A1 state above the a˜ 1A1 state at the a˜ state equilibrium geometry is determined to be 35 984 cm-1 (36 652 cm-1 in ref 2), in disagreement with the region in which the unassigned lines in Table 3 of ref 6 are observed (27 000-30 000 cm-1). We determine ab initio that in the c˜ 1A1 state, CH2 is almost linear, with an equilibrium angle of 172.7° and a barrier to linearity of only 6 cm-1, so that the bending potential energy function has a rather unusual shape. The cosine expansion (eq 4), which we use to represent the bending potential, appears to be poorly suited to represent a function of this type. In Figure 1 we plot the functions n
V(n)(Fj) ) ∑ f(i) j )i 0 (cos Fe - cos F
(5)
i)2
for n ) 2, 3, ..., 5 as dashed lines. In this notation, the actual bending potential for c˜ 1A1 CH2, V0(Fj) ) V(6)(Fj). This function is given as a full curve in Figure 1. It is seen that for c˜ 1A1 CH2, the expansion given by eq 4 is extremely poorly convergent, as reflected by the numerically large values for the f(i) 0 parameters given in Table 2. For potential energy functions with a larger barrier to linearity, this expansion is generally much
18092 J. Phys. Chem., Vol. 100, No. 46, 1996
Bunker et al. integration technique uses as input a table of V0(F) values obtained at a grid of equally spaced F values. However, for c˜ 1A CH the very large numerical values of the f(i) parameters 1 2 0 cause the (r1,r3)-dependent terms resulting from the F f Fj transformation to be extremely large, and these terms cause strong interactions between the basis functions used to construct the MORBID matrix representation of the rotation-vibration Hamiltonian. Owing to these interactions, we must use very large basis sets in order to obtain converged rotation-vibration energies for c˜ 1A1 CH2. We hope that the theoretical results reported here will facilitate the spectroscopic characterization of the c˜ 1A1 electronic state of CH2. Acknowledgment. This research was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Fundamental Interaction Branch, Grant No. DE-FG05-94ER14428, by the Deutsche Forschungsgemeinschaft (through Forschergruppe Grant Bu 152/ 12-5), and by the Fonds der Chemischen Industrie. References and Notes
Figure 2. Bending potential V0(Fj) (eq 4, full curve) and the functions V(n)(Fj) (eq 5, dashed curves) for X ˜ 3B1 CH2.
better convergent, as illustrated by Figure 2, which shows the bending potential V0(Fj) ) V(6)(Fj) (full curve) and the functions ˜ 3B1 CH2.14 In this V(n)(Fj) (dashed curves) given by eq 5 for X -1 state, the barrier to linearity is 1916 cm . The parameter values used to calculate the functions shown in Figure 2 are obtained from the “Fitted” column in Table II of ref 14. The kinetic energy in the MORBID Hamiltonian is not expressed in terms of Fj, but in terms of an angle F, which is almost, but not quite, equal to Fj for molecular configurations with (r1,r3) * (r1e,r3e). Consequently, the potential energy function V(∆r1,∆r3,Fj) (eq 1) must be transformed to depend on ∆r1, ∆r3, and F. This is done by means of eqs 36 and 37 of ref 3. The transformed potential contains the term V0(F) together with a number of (r1,r3)-dependent terms proportional to the expansion coefficients f(i) 0 of eq 4. In the variational MORBID calculation, the term V0(F) is initially used to generate the bending basis functions in a Numerov-Cooley integration17 (see section V of ref 3). At this stage of the calculation, the poor convergence of the expansion in eq 4 is of no importance for the efficiency of the computation since the Numerov-Cooley
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