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On the Theoretical Calculation of Vibrational Frequencies and Intensities of Polyatomic Molecules; H3 +, H2D+, HD2 +, and D3 +
1.For example, E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations (McGraw‐Hill Book Company, Inc., New York, 1955).
2.F. O. Ellison, N. T. Huff, and J. C. Patel, J. Am. Chem. Soc. 85, 3544 (1963).
3.One might argue that the calculation could be simplified by first reducing the order of the secular equation by eliminating translation and rotation; that is, five or six relations between displacements which cause the center of mass to remain stationary and the angular momentum to equal zero, are imposed initially (see Ref. 1, pp. 12–14, 22–25). The conditions consequent on the would then be obtained through Eqs. (4). This procedure, however, is unnecessarily complicated for machine computation. The symmetry reduced secular equations will contain at most two zero roots in any given symmetry block (except for molecules belonging to Groups or ); the efficiency of a computer secular equation solver is not lowered seriously by an increase from n to dimensions (e.g., for benzene, the secular equations reduce to eight independent and two independent equations if translation and rotation are included; they reduce to three independent five independent one independent and one independent equations if translation and rotation are removed).
4.If two or more translational and/or rotational modes belong to the same species, then there will be a corresponding number of zero roots, and the associated normal coordinates may be considered as accidentally degenerate. Ordinarily, one will not be interested in calculating the which relate these particular normal coordinates to the symmetry coordinates [Eq. (11)]. However, in the method for predicting absolute intensities proposed in the next section, these transformation coefficients are required. A direct means for accomplishing this is described in Appendix III.
5.See Ref. 1, p. 164; also Ref. 6.
6.B. Crawford, Jr., J. Chem. Phys. 29, 1042 (1958);
6.I. M. Mills and D. H. Whiffen, J. Chem. Phys. 30, 1619 (1959); , J. Chem. Phys.
6.for a review of intensities and compilation of common units, see L. A. Gribov and V. N. Smirnov, Soviet Phys.‐Uspekhi 4, 919 (1962)
6.[L. A. Gribov and V. N. Smirnov, Usp. Fiz. Nauk SSSR 75, 527 (1961)].
7.The special theory for predicting intensities of vibrational and rotational spectra of charged diatomic molecules has been developed previously by F. O. Ellison, J. Chem. Phys. 36, 478 (1962).
7.Analogous expressions for charged polyatomic molecules can be derived rather easily. If the molecules possesses zero net electrostatic charge, a unique origin for the and in Eq. (22) must be chosen, but is independent of this choice of origin. [For pure electronic transition probabilities, the transition moments are independent of the choice of origin for the dipole moment operator regardless of the net electrostatic charge associated with the molecule; see H. Shull and F. O. Ellison, J. Chem. Phys. 19, 1610 (1951).]
8.D. G. Williams, W. B. Person, and B. Crawford, Jr., J. Chem. Phys. 23, 179 (1955).
9.R. E. Christoffersen, S. Hagstrom, and F. Prosser, J. Chem. Phys. 40, 236 (1964).
10.R. E. Christoffersen, “A Configuration Iteration Study of the Ground State of the Molecule,” Indiana University Theoretical Chemistry Laboratory Report, December 1963.
11.H. Conroy, J. Chem. Phys. 40, 603 (1964).
12.W. D. Jones and W. T. Simpson, J. Chem. Phys. 32, 1747 (1960).
13.(a) Let be the coordinates of nucleus α relative to the center of mass for the molecule in its equilibrium configuration. For an arbitrary set of Cartesian displacement coordinates the new x coordinate of nucleus α, relative to the old center of mass, is similar expressions hold for and The internuclear distance in the displaced configuration is given by which may be computed from the and
13.(b) If b and/or c do not equal zero, one may obtain an improved set of nuclear equilibrium coordinates (see Footnote 13a) by differentiating Eq. (33) with respect to and with respect to setting the results equal to zero, and solving for and One can transform to the corresponding using the inverse of Eqs. (4), and then to the Cartesian displacement coordinates which serve as corrections to the originally assumed
14.One hartree unit where μ is taken as the rest mass of an electron; one bohr unit see H. Shull and G. G. Hall, Nature 184, 1559 (1959).
15.Since infrared frequencies are usually large numbers, the corresponding wavenumbers are most often quoted. If we define and Eqs. (8) and (9) may be written with and replacing and λ, respectively. To obtain the from the values calculated theoretically (in units of hartree per atomic mass ), simply apply the conversion factor
16.All constants and conversion factors calculated using physical constants given in Phys. Today 17, No. 2, 48 (1964).
17.See Footnote 13a. Since the only displacements considered in calculating the are normal vibrational displacement coordinates, the center of mass necessarily will be stationary.
18.In the more general case, integrals of the type will not be zero necessarily. Such could be evaluated using formulas given by Hamilton, J. Chem. Phys. 26, 1018 (1957).
19.Again, in the more general case, integrals of the types (in which p and are not identical 1s orbitals on the two centers) will not necessarily be zero. In such cases, the relative orientations of the axes used for specifying the atomic orbitals on the two centers as well as of the axes could be taken into account to yield basic dipole moment integrals formulated by Hamilton.18
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