J. Phys. Chem. 1985,89, 3876-3879
3876
Variational Calculations of Rotational-Vibrational Energy Levels of Formaldehyde
'A,
Barbel Maessen and Max Wolfsberg* Department of Chemistry, University of California, Iruine, California 92717 (Received: January 7, 1985)
Variational calculations of rotational-vibrational energy levels are carried out on H,CO and D2C0 for J I i0 within the framework of the Watson Hamiltonian. The potential function, written in valence displacement coordinates, is one used previously in calculations for J = 0. The basis functions are products of vibrational functions and symmetric top rotor functions, where the vibrational functions diagonalize the rotationless ( J = 0) Hamiltonian. Integrations over normal coordinates are carried out numerically by Gauss-Hermite quadrature. Calculations for the vibrational ground state are compared with experimental observations. The importance of Coriolis interactions on the energy levels and wave functions is demonstrated.
I. Introduction Variational calculations of the lower lying vibrational states ( J = 0) of formaldehyde in its electronic ground state have been reported for different force fields.' It was found that a modification of the quartic force field of Tanaka and Machida,2 which was derived from experiment within the framework of second-order perturbation theory, yields reasonable agreement between calculated vibrational energies and observed spectra. The modification consists of the replacement of the terms in this force field which involve stretching displacements Ar by terms involving Ar/r such that the original force field and the new one agree up through quartic terms. In the present study, the calculations are extended to describe the rotational-vibrational spectrum of formaldehyde. These calculations are based on the Watson form of the full Hamiltonian in normal coordinate^,^ as were the calculations for J = 0.' The basis functions are products of vibrational functions and symmetric top functions where the vibrational functions xu are functions which diagonalize the Hamiltonian for J = 0. The calculations have been carried out for J 5 10. Of interest is comparison between theory and available experimental data. Also of interest is the calculated effect of vibration on the rotational spectrum and the mixing of different vibrational functions xu into the same rotational-vibrational wave function. Details of the calculations are given in section 11, and the results are presented in section 111. 11. Technique A . General Considerations. The Watson3 form of the molecular
rotational-vibrational Hamiltonian is 1 H = -cPk2 2 k
1
+ +,,(II, a .B
- a,)(II,
- a& -
h2
--ZPaa +V 8 , (1)
Here Pk is the momentum conjugate to the normal coordinate Qk, element of the inverse of the effective moment of inertia tensor, II, is the operator for the Cartesian component (molecule-fixed axes) of the total angular momentum, and a, is the corresponding operator for vibrational angular momentum
the normal coordinates, and of the symmetric top rotor functions SIKM,which are functions of the three Euler angles. Both J and M are good quantum numbers for the molecule, and the M quantum number produces a ( 2 J + 1)-fold degeneracy. The M quantum number is consequently set equal to zero, and the symmetric rotor functions are needed only for M = 0. These functions, which are functions of only two Euler angles, are designated SIK(t9,6)so that the basis functions are *~JK
=
x J Q SJK(~J) ~
(3)
The functions x,(Qk) are taken to be a set of functions which diagonalize the Hamiltonian operator for the case J = 0 (Le., for the case when all terms involving, II, are set equal to zero). The basis functions for the calculation of the pure vibrational ( J = 0) energy levels are taken to be products of six harmonic oscillator wave functions. The method of solution is an extension of that of Whitehead and Handf so that the integrals over normal coordinates are evaluated by Gauss-Hermite integration. If a basis functions are used in the J = 0 problem, one obtains, of course, a functions xu(&). A truncated set of these xu(&) functions in the basis functions \EoJKof eq 3 is expected to lead to much better convergence of the problem with J # 0 than would an equal number of products of six harmonic oscillator functions. In the calculations here with J # 0, the relevant matrix elements over the normal coordinates are evaluated by GaussHermite integration as in the problem for J = 0. The integrals over Euler angles are evaluated analytically. For the latter integrations, the angular momentum matrix elements of King, Hainer, and CrossS are used. As already noted, J is a good quantum number while K is not; however, in the construction of the matrix elements of the full Hamiltonian with respect to the basis set of q , 3 , (v'JKIHlvJK), there are very specific selection rules with respect to K . The nonzero matrix elements (with h = 1) follow for K' = K
pa@is an
= XEIQ~PI k,l
(2)
where E,is a Coriolis coefficient. In the equilibrium configuration, the molecule-fixed axes here correspond to principal axes. Vis the intramolecular potential energy expression. The different terms have been previously d i s c ~ s s e d . ' ~ ~ * ~ The basis functions used for the calculations with J # 0 are products of vibrational functions xu(Qk),which are functions of (1) B. Maessen and M. Wolfsberg, J . Phys. Chem., 88, 6420 (1984). (2) Y. Tanaka and K. Machida, J. Mol. Specrrosc., 64,429 (1977). (3) J. K. G. Watson, Mol. Phys., 15, 479 (1968). (4) R. J. Whitehead and N. C. Handy, J . Mol. Specrrosc., 55, 356 (1975); 59, 459 (1976).
0022-3654/85/2089-3876$01.50/0
U
i[+2(PXZ),&K
f
1)
f
(cP,x*a)"~"l) a
for K' = K
f
2: , 4 4 Y 2 ( ( P Y Y ) d "-
(PXX)",")
?=
i(P*Y)"%I
(4)
+
where A = ' / , [ J ( J 1) - K ( K f l ) ] ' / * , B = ' / 2 [ J ( J+ 1 ) - ( K f 1)(K f 2)]'/2, and ( ~ x x ) , , v = ( u j p X X ~ v )Here . E, is the ei(5) G. W. King, R. M. Hainer, and P. C. Cross, J . Chem. Phys., 11, 27 (1943); P. C. Cross, R. M. Hainer, and G. W. King, J . Chem. Phys., 12,210 (1944).
0 1985 American Chemical Society
Rotational-Vibrational Energy Levels of Formaldehyde TABLE I: Vibrational Energy Levels (cm-I)' for Formaldehyde ( J = 0) UCalCd
no.
assignt 1167.3 1249.1 1500.2 1746.0 2328.9 2655.5 2719.2 2782.5 2843.3
548 BFb 5796.0 1162.0 1250.4 1500.1 1746.7 2307.6 2400.6 2490.9 2656.9 2748.0 2792.9 2862.3
The Journal of Physical Chemistry, Vol. 89, No. 18, 1985 3877 TABLE 11: Symmetries in Point Group C b of the Operators of Eq 4"
operators
symmetry
Pyyr Prr
AI
PXXl
48 BF 5818.0 1175.8 1259.2 1510.1 1761.1 2334.5 2437.4 2503.5 2674.6 2764.0 2815.1 2875.6
'Energies given with respect to ground state, the energy of which is given with respect to potential minimum. bSee ref 1. CGroundstate. genvalue of the problem for J = 0 which corresponds to the eigenvector xc. The selection rules on the matrix elements over normal coordinates depend on the axes choices and will be discussed subsequently. The fact that these matrix elements are not diagonal in u and u'leads to mixing of different xu functions into a given eigenfunction obtained for the problem with J # 0. The matrix elements containing the vibrational angular momentum operator A, permit the mixing of xu functions corresponding to different symmetries of the point group Czu. Such mixing is referred to as Coriolis mixing, even if it is brought about by matrix elements not containing A,. Mixing of x i s of the same symmetry by matrix elements with A, symmetry is referred to as centrifugal mixing by analogy to the same type of mixing brought about by the centrifugal stretching term in diatomic molecules. In calculations of the rotational-vibrational states of water, it is found that both types of mixing are important. The calculations here were carried out with the complex symmetric top wave functions SjK of eq 3.5 The HzCOmolecule was oriented with the z axis of quantization being the 2-fold molecular axis. For this orientation, mixing of +uJK functions is permitted as follows: (1) AI and A2 xu functions with K even in SjK, B, and Bz xu functions with K odd and (2) A I and Az xu functions with K odd in SjK, B , and B2 xu functions with K even, so that the secular equations can be separated into two smaller eigenvalue problems. With the use of the real Wang-type symmetric top function^,^ it is possible to introduce further symmetry factoring into the secular equations of the variational problem to reduce the computational complexities. B. Details of the Calculation. The choice of the parameters of the present calculations is based on the experience of earlier variational calculations on vibrational states of formaldehyde. The force field I/ (and geometry) here, as already noted, corresponds to the A r / r reexpansion of the Tanaka-Machida perturbation theory force field.2 The parameters are given in ref 1. The vibrational basis set for the calculations with J = 0 is based on an analysis of the Ybestncalculation by the present authors' in which 548 basis functions (BF are products of six harmonic normal-coordinate oscillator functions, one for each normal mode) were used. The point group symmetry designations of these BF follows: 196 A,, 165 B2, 108 B1, 79 AZ. Such a large number of basis functions would make the calculation of the vibrational matrix elements needed for the rotational-vibrational interaction very expensive in computer time. The basis function set for the construction of the xu functions here is therefore reduced to 48 BF as follows. Only those BF of the 548 BF calculation are used for the smaller calculation which have coefficients of 0.1 or larger in the normalized eigenvectors of the vibrational ground state or of any vibrational state with energy equal to or less than 3000 cm-' above the ground state (which corresponds to the 12 lowest states); this analysis yields 48 BF (20 AI, 14 B2, 12 B,, 2 Az). The vibrational energies obtained for the lower lying vibrational states with 48 BF are compared with those obtained in the larger
*y
B2 BI
*z
A2
Efiuxr.9 Pyz, r x
CPaJTa, P x z , ZPmra, P X J n
'Choice of axes z = a, y = b, x = c. The nonzero Coriolis constants are in terms of the symmetries of Qd and Q, - (a,Qd,Q,): ( z $ h b ) , Cy,Al,B1), (x,Al,B2).
calculation and with experiment in Table I. The agreements are reasonable. The designation of the states is the usual spectroscopic one, determined by the BF which makes the largest contribution to the corresponding eigenvector. The 12 lowest functions resulting from the 48 BF calculation for J = 0 are used to construct the basis of eq 3 for the calculations here with J # 0. The symmetry designations and corresponding spectroscopic designations for these x i s are shown in Table I. The use of 12 functions has been shown to yield reasonable convergence for the lower vibrational states with J 5 10 in the case of water. Formaldehyde is a near prolate asymmetric rotor (at equilibrium I,,
