2878
J. Phys. Chem. 1981, 85, 2878-2881
A Theoretical Determination of the Overtone and Combination Band Intensities for v3 and v4 of Methane Kerin Scanlon, Robert A. Eades, David A. Dixon,*+ and John Overend” Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: July 16, 198 1)
A number of derivatives of the dipole moment function for CHI have been calculated from ab initio molecular orbital theory in the finite difference approximation. All calculations were carried out in normal coordinates with an experimental force field and geometry. A near Hartree-Fock basis set of the form (11,7,2/5,1)/[7,6,2/3,1] was employed. The calculated harmonic frequencies in cm-’ are 3050 (a1),1679 (OZ), 3139 (as),and 1480 (a4). The calculated values of the intensities in km/mol are 108.5 (v3) and 26.2 (v4) which correspond to dipole moment derivatives in e amu1J2of ap,/aQa, = -0.1964 and ap,/aQ6 = -0.1007. Second derivatives of the dipole moment function for certain overtone and combination bands of v3 and v4 have been calculated in units of e amu-l k1 to be aZp,/aQlaQ3, = -0.1928, a2px/aQ3yaQ3z= -0.4162, aZp,/ag,a~,, = 0.1951, and a2p,/a~4ya~4z = 0.2192. These results demonstrate that the electrical anharmonicity corrections are of comparable magnitude with the mechanical anharmonicity corrections. Furthermore, the second derivatives of the dipole moment function for CHI are essentially independent of substitution of 13C for IzC.
Introduction Infrared intensitites of many simple molecules have been extensively studied experimentally and have been interpreted to obtain derivatives in the molecular electric dipole moment with respect to defined intramolecular distortions c0ordinates.l As in a number of other techniques, e.g., X-ray crystallography, the experimental measurements yield magnitudes but not phases and the sign of the dipole-moment derivative is not determined from the experimental intensity. The sign of the derivative can be determined from theory and, in recent years, first derivatives of the molecular dipole moment have been obtained from wave functions calculated from ab initio molecular orbital t h e ~ r y and ~ - ~have given calculated infrared intensities of a substantial number of fundamental bands in reasonably good quantitative agreement with experiment. Dipole derivatives have been calculated by both the finite difference appro xi ma ti or^^-^ and the gradient methodS5 The intensity of an infrared fundamental band is given by6 8r3N0 1 A , (km/mol) = -v?=l3hc 2a=x,y,z where No is Avogadro’s number, h is Planck‘s constant, c is the speed of light, and vi is the observed wavenumber of the u . = 0 uj = 1 transition. The quantity (dpa/dqj) is the derivative of the a component (a = x,y,z) in the laboratory-fixed frame of the molecular dipole moment with respect to the jth dimensionless normal coordinate. Similar expressions have been derived for the intensities of binary overtone and combination bands’ through second-order perturbation theory:
-
e-zgwj) ( 2 ) +Camille and Henry Dreyfus Teacher-Scholar (1978-83). A. P. Sloan Foundation Fellow (1977-81). DuPont Young Faculty Grantee (1978). 0022-365418 1 120a5-287a$o12 5 1 0
where /3 = l / k T and zj is the vibrational partition function for the jth normal mode. The quantities Sjkl and Sik1are the coefficients of the operators in the first contact transformation. They are functions of the cubic normal-coordinate force constants and the harmonic frequencies; explicit expressions have been given by Amat, Goldsmith, and Nielsen.8 From eq 2 and 3 it is obvious that the intensities of binary overtone and combination bands depend on the magnitudes and the signs of both the first and second derivatives of the dipole moment. A report9 that the intensity of the 2v3 band of methane was apparently significantly reduced when I3C was substituted for 12C aroused our interest in the overtone and combination bands of methane. The contributions from (1) (a) W. B. Person and D. Steele in “Molecular Spectroscopy”, Vol. 2, R. F. Barrow, D. A. Long, and D. J. Millen, Ed., Chemical Society, London, 1974, p 357; (b) D. Steele in “Molecular Spectroscopy”, Vol. 5, R. F. Barrow, D. A. Long, and J. Sheridan, Ed., Chemical Society, London, 1977, p 106; (c) L. A. Pugh and K. N. Rao in “Molecular Spectroscopy: Modern Research”, Vol. 11, K. N. Rao, Ed., Academic Press, New York, 1976, p 165. (2) (a) J. H. G. Bode and W. M. A. Smit, J . Phys. Chem., 84, 198 (1980); (b) W. M. A. Smit and T. van Dam, J . Chem. Phys., 72, 3658 (1980). (3) P. Pulay and W. Meyer, J . Chem. Phys., 57, 3337 (1972). (4) W. Meyer and P. Pulay, J. Chem. Phys., 56, 2109 (1972). (5) (a) A. Komornicki and J. W. McLiver, Jr., J. Chem. Phys., 70,2014 (1979); (b) A. Komornicki and R. L. Jaffe, ibid., 71, 2150 (1979). (6) J. Overend in “Infrared Spectroscopy and Molecular Structure”, M. M. Davies, Ed., Elsevier, Amsterdam, 1963, Chapter X. (7) (a) C. Secroun, A. Barbe, and P. Jouve, J . Mol. Spectrosc., 45, 1 (1973); (b) S. J. Yao, Ph.D. Thesis, University of Minnesota, 1966; (c) S. J. Yao and J. Overend, Spectrochim. Acta, Part A , 32,1059 (1976); (d) J. Overend, “Proceedings of Nato Conference”, Italy, 1977, in press. (8) (a) G. Amat, M. Goldsmith, and H. H. Nielson, J. Chem. Phys., 27, 838 (1957); (b) M. Goldsmith, G. Amat, and H. H. Nielson, ibid., 24,1178 (1956). (9) (a) K. Fox, G. W. Halsey, S. J. Daunt, W. E. Blass, and D. E. Jennings, J . Chem. Phys., 72,4657 (1980); (b) P. Varanashi, J. Quant. Spectrosc. Radiat. Transfer, 25, 319 (1981). 1981 American Chemical Society
The Journal of Physical Chemistty, Vol. 85,
Letters
mechanical anharmonicity to the intensities of the overtone and combination bands were estimated by use of normal-coordinate force constants derived from an assumed force field based on the quadratic field of Jones and McDowell'O augmented with principal cubic and quartic CH stretching force constants derived from the assumed Morse parameter, a = 2.0 A-', cf. ref 11 (see Table I). The values of dp/dq3 and dp/dq4 were based on the observed intensities of the v3 and v4 fundamentals.12 From these data we were able to calculate the contributions from mechanical anharmonicity to the intensities of 2v3 and v1 v3 of CH4. The results of this calculation were, first, that the contributions from mechanical anharmonicity to 2v3 were the same sign and similar in magnitude to the contributions to the transition moment of v1 v3 and, second, that the off-diagonal element of the second-order Hamiltonian coupling the 2v3 and v1 + v3 states was large compared with their calculated separation. These results led to the prediction of a relatively intense band at -5858 cm-l and a much weaker one -6016 cm-'. The experimental facts in W H 4 are just the reverse, i.e., the band observed at 6005 cm-l is much more intense than that at 5861 cm-'.13 To resolve this apparent inconsistency we postulated that the second derivatives of the dipole moment, (d2p/ dqS2)and d2p/dqldq3), made significant contributions to the intensities of 2v3 and v1 + v3 and, by requiring that the calculated intensities of 2v3 and v1 + v3 match the observed ones, we were able to determine sets of empirical values for the two second derivatives, each set corresponding to different assumed signs for the experimental dipole transition moments. Since the empirical values, especially that of the mixed derivative, were larger than we expected, we decided to calculate the second derivatives of the molecular dipole moment of methane from ab initio wave functions. This allowed us to predict the magnitude and the sign of the second derivatives and, thus, to make the proper choice of phase for the dipole moment matrix elements evaluated from the experimental measurements. We have also calculated the second derivatives for the overtone band, 2v4, and combination band, v1 + v4, for the bending mode. These results represent the first good calculations of the second derivative of the dipole moment function for a polyatomic molecule.
+
+
Calculations All ab initio calculations were performed with the HONDO: Version 5 program14 on either a CDC Cyber 74 or a DEC VAX-111780 computer. The force field and intensity calculations were performed on programs developed at the University of Minne~0ta.l~All calculations were carried out by use of experimental geometry16for CHI (rcH = 1.094 A) although calculations with the optimum theoretical geometry were in good agreement.17 A near (10) L. H. Jones and R. S. McDowell,J. Mol. Spectrosc., 3,632 (1959). (11) (a) I. Suzuki and J. Overend, Spectrochim. Acta, in press; (b) A. C. Jeannette, 11, C. Marcott, and J. Overend, J. Chern. Phys., 68, 2076 (1978). (12) We employ the average values for the fundamental intensities given by 50%. This is typical of near Hartree-Fock values where we expect to find agreement within a factor of ~ w o . ~ - ~ The frequencies derived from the Meyer and Pulay theoretical force field are in good agreement with ours which were obtained by use of an experimental force field. Their values of the derivatives are similar to ours with the
-
(22)In unpublished work on NHs, intensities were calculated by use of different force fields and these size displacements. The sum of the fundamental intensities was found to remain constant demonstrating that the displacements are small enough to yield the derivative at the chosen geometry. (23)A normal coordinate calculation was performed by use of the ab initio force field and theoretical optimum geometry given in ref 4. The dipole moment derivatives with respect to normal coordinates and the associated intensities were obtained by use of the force field and geometry of ref 4 to transform the symmetry coordinate derivatives to normal coordinate derivatives. A similar calculation was performed with the experimental force field and geometry given in ref 10 and the results were found to be essentially the same.
Letters
first derivative for v3 being 35% larger than the experimental values and the first derivative for u4 being 9% below the experimental quantity. Thus reasonable agreement between the two sets of calculations is found which is to be expected since similar geometries and near Hartree-Fock basis sets were used in both calculations. The slight discrepancies in the values of the derivatives could easily be due to the use of different force fields to determine the normal coordinate motions. The calculated second derivatives are given in Table I11 together with the first derivatives obtained from the second derivative calculation. The various experimental estimates for the second derivatives, assuming no vibrational resonance, vibrational resonance, and vibrational plus Fermi resonance, are given Table IV. These latter quantities were obtained after subtracting the mechanical anharmonicity contributions from the experimental intensities. There are actually four possible sign choices for the dipole moment matrix elements for the first set of experimental derivatives but the combinations of plus and minus signs give essentially zero intensity for both modes and, therefore, only two sets need be considered. The theoretical values for the second derivative (both sign and magnitude) show that for 2u3 and u1 + v3 both dipole matriix elements should be less than zero in order for the calculated intensities to agree with the experimental intensities. Good agreement between theory and experiment is found with the largest error occurring for 2v3. If resonance effects are included, the best choice of sign for the matrix elements is [ A ( ~ v ~ )>] 0' /and ~ [A(vl u3)I1/' < 0. Better agreement is found, however, if the resonance effects are not included. The second derivatives for the bending mode overtone and combination bands are found to be positive and to have essentially the same value of -0.2 e amu-l A-1. Only the The value intensity of v1 + u4 is known e~perirnentally.~~ of the second derivative derived from experiment for u1 + u4 is in excellent agreement with the theoretical value if the choice of sign for the matrix element [A(vl u4)]1/2is ) ] ' be / ~made from positive. No choice of sign for [ A ( ~ U ~can experiment since this intensity has not yet been measured. We also report the second derivatives for 12CD4and 13CH4 in Table V. A check on the internal consistency of the calculations is to determine the first derivatives SpJSQ3x and Spxf SQ, from the second derivative displacements. As shown in Table 111, all values are in excellent agreement with the values calculated from the first derivative displacements (see Table 11)and are independent of the second derivative (overtone or combination) from which they were calculated.
+
+
Discussion The Hartree-Fock values of the various derivatives calculated for CH4 are in reasonable agreement with the experimental quantities. For CH4,for small displacements near the equilibrium geometry, the Hartree-Fock values should yield adequate values for the dipole moment derivatives. Of course as the bond distance increases, the derivatives will show a larger error since the Hartree-Fock wave functions force the molecule to dissociate to ions rather than to neutral radicals. For large displacements, the calculations should be performed with correlated wave functions. For an AH molecule with no permanent dipole, the derivative of the dipole moment function in the stretching coordinate should be too large. Hartree-Fock wave functions usually give dipole moments that are too large as compared to experiment and, thus, for a value of (24)J. Vincent-Geisse, C. R. Acad. Sci. Paris, 239,251 (1954).
The Journal of Physical Chemistry, Vol. 85, No. 20, 1981 2881
Letters
TABLE 111: Theoretical Values for t h e Second Derivatives of
"TH, best expta,c
derivative
value"
a'~,/aQ,aQ~~ a ' ~ , / a Q , ~ a Q,+
-0.1928 -0.4162 0.1951 0.2192
aZ~x/aQ,aQ4x aaP,/aQ,,aQ,,
value of firstb derivative -0.1964 -0.1964 0.1008 0.1008
n o resonance (-,- ;+ I -0.1956 -0.2782 0.1789
resonance
(+,-I -0.0948 -0.2719
" Second derivative in e a m u - ' A-l. First derivative evaluated from the second derivative calculation in e amu-lt2. possible second derivatives for 2v, and v l + v g are shown in Table IV. TABLE IV: Experimental Estimates of t h e Second Derivatives of t h e Dipole Moment Function for 'TH, derived second derivatives signsa
[A(2v3)1"'
[ A ( v ,+ v , ) ] " ~
a 'P,I aQ,aQ3,
azp,/
aQ,,aQ3+
No Resonance
+
-0,1095 -0.1956
-0.0370 -0.2782
Vibrational Resonance + -0.0355 -0.0948 -0,2103 + -0.2696
-0.2092 -0.2719 -0.0433 -0.1060
t
-
-
+ +
-
+ +
Vibrational
.-
-
+ + -
+
Fermi Resonance -0.0285 -0.0957 -0,2094 -0.2766
-0.2455 -0.3020 -0.0133 -0.0697
a Sign of the square r o o t of the intensity. Values of the second derivatives obtained b y subtracting o u t t h e ef. fects of mechanical aharmonicity in units of e amu-' A - ' .
TABLE V : Theoretical Second Derivatives for Different Methane Isotopes (e a m u - ' A - ' ) derivative
"CH,
WH,
TD,
a2pr/aQlaQ3,
-0.1928 -0.4162 0.1951 0.2192
-0.1921 -0.4133 0.1944 0.2177
-0.1007 -0.2270 0.1019 0.1195
a2px.aQ3yQ3z
aapx/aQlaQ4x aZp,/aQ4yaQ4z
0, the derivatives will be too large. This is due to an overestimation of the electron density between the bonds due to an overweighting of the ionic contribution^.^^ The values for the second derivatives are in good agreement with the predicted experimental values. Al-
bo =
(25) It is tempting to speculate that the derivative of the dipole moment function for the bend is too low for the same reason, Le., there is too little variation in charge density on bending because there is too much density in the bonding region. This is not borne out by calculations on other AX4 molecules, however.
All
though better agreement is found if resonance effects are excluded, we cannot definitively exclude them. However, we note that the resonance effects must be included in order to obtain the correct magnitude of the frequencies. The calculated values demonstrate that the electrical anharmonicity term is of the same magnitude as the mechanical anharmonicity term and cannot be excluded in a rigorous calulation of overtone and combination band intensities. The second derivatives of the dipole moment function show essentially no change on substitution of 13C for lZCin CH4 as shown in Table V. A significant effect is found for substitution of D for H to give WD4. However, this effect is the same as that observed experimentally and is simply due to the large change in reduced mass upon substitution of D and due to the large motion of the hydrogen atom relative to the carbon atom in the v3 and v4 vibrations. The present analysis of the intensities of the 2v3 and v1 + v3 bands of methane gives a picture which is internally consistent for 12CH4and 12CD4. It does not explain the lowering of the intensity of the 2v3 band in I3CH4which is expected from the intensities of single rotation-vibration lines.g Quite frankly this result still puzzles us. In conjunction with mechanical anharmonicity terms derived from experimental fundamental intensity measurements and anharmonic force fields, it is now possible to make reasonable estimates of higher order intensities (overtone and combination bands) by use of second derivatives derived from ab initio wave functions. These quantities will have significant applications in laser chemistry and in molecular spectroscopy. Further studies on other AX4 molecules and ions including the details of the resonance calculations for CH4 will be reported in subsequent publications.
Acknowledgment. David A. Dixon acknowledges the National Science Foundation (Grant CHE-7905985) for partial support of this work. John Overend acknowledges partial support from the donors of the Petroleum Research Fund, administered by the American Chemical Society. Kerin Scanlon acknowledges the support of a Louise T. Dosdall Pre-Doctoral Fellowship from the University of Minnesota Graduate School (1980-81).
