Dyes: quantum chemical calculation of electronic spectra

A Appendix

In the chapter “Calculation of the Shape of Absorption Bands in Molecular Electronic Spectra” the applicability of the diatomic FCP to polyatomic molecules was discussed. In their basic approach for polyatomic molecules Herzberg and Teller have shown that vibrations can appear as a progression only if the symmetry is the same in the excited electronic state as in the ground state; i. e. an intense symmetry allowed electronic transition can couple only with symmetric vibrations, which change the size but not the symmetry of the molecule, just as is the case for the symmetric valence vibration in a diatomic molecule.

This lays the initial foundation anti-symmetric vibrations cannot couple with an electric dipole allowed electronic transition. In addition it is fundamental knowledge the anti-symmetric vibrations are IR active and not Raman active.

In the cases of homonuclear as well as in heteronuclear diatomic molecules the symmetric valence vibration changes the polarizability, so vibrations in both molecule types are Raman active. This point is also clear.

In many text books one can read the following or similar selection rules for IR active transitions:

“The condition for a normal vibration j to be IR active is a change in molecular dipole moment μ during vibration.”[A1]

“The dipole moment μ is zero for a homonuclear diatomic molecule, resulting in R_v = 0 and all vibrational transitions being forbidden. For a heteronuclear diatomic molecule μ is non-zero and varies with x.”[A2]

“The gross selection rule for a change in vibrational state brought about by absorption or emission of radiation is that the electric dipole moment of the molecule must change when the atoms are displaced relative to one another. Such vibrations are said to be infrared active.”[A3]

“In order for energy to be transferred from the IR photon to the molecule via absorption, the molecular vibration must cause a change in the dipole moment of the molecule. This is the familiar selection rule for IR spectroscopy, which requires a change in the dipole moment during the vibration to be IR active.

Homonuclear diatomic molecules such as H₂, N₂, and O₂ have no dipole moment and are IR inactive (but Raman active) while heteronuclear diatomic molecules such as HCl, NO, and CO do have dipole moments and have IR active vibrations.”[A4]

Due to these rules the symmetric valence vibration in heteronuclear diatomic molecules shall be IR active because the change in molecular electric dipole moment during vibration Thus, often “seemingly suitable” IR vibration frequencies in polyatomic molecules are used to explain the vibrational fine structure in electronic spectra. However, is this justified?

Therefore, it is important to clarify, is the symmetric valence vibration in diatomic molecules IR and/or Raman active?

From the experimental point of view all these statements are invalid. Neither in the high-resolution rotation-vibration absorption spectra of HCl nor CO or other heteronuclear diatomic molecules is there an IR active normal vibration. This message is especially surprising, because the IR spectrum of HCl is acquired and analyzed already in undergraduate physical chemistry laboratories, yet in the experimental spectra there is no hint of a vibration absorption band [A5, A6].

Due to the cited vibrational selection rule, the transition v = 0 → v´ = 1 shall be IR active. But where is the absorption band of this vibrational transition?

In molecules there are transitions between separate rotational, vibrational and electronic states and the lower energy transitions can be associated with the excitation of higher energy transitions. The rotation-vibration absorption spectrum of HCl shows two branches, the R (v = 0 → v´= 1; ΔJ = +1) and P (v = 0 → v´= 1; ΔJ = –1) branches of the rotational transitions, whereas the absorption band for the vibrational transition v = 0 → v´= 1 is absent (Table 4). It is expected that the frequency of the “missing vibrational absorption band” v = 0 → v´ = 1 is between the R(0) (v = 0, J = 0 → v´ = 1, J´ = 1) and P(1) (v = 0, J = 1 → v´ = 1, J´ = 0) rotational transitions (v - vibrational quantum number, J - rotational quantum number).

The spacing between rotational absorption bands varies slowly as a function of the vibrational states. Assuming constant spacing between the R(0) and the P(1) transitions (Table 4) the midpoint has a value of 2,885 cm^–1. This midpoint frequency corresponds with the frequency of the “missing vibrational absorption band” v = 0 → v´ = 1, which is obviously a non IR active vibration.

The rotation-vibration scattering spectrum (Raman spectrum) of HCl was first time measured by R. W. Wood [A7]. He determined the frequency of the “missing vibrational absorption band” to be 3.4645 μm (2,886 cm^–1) [A7].

So, since 1929 it has been known that the frequency of the “missing vibrational absorption band” in the rotation-vibration absorption spectrum of HCl can be observed in rotation-vibration scattering spectrum.

Also in the rotation-vibration absorption spectrum of CO the band of the fundamental CO vibrational transition is absent (Table 5) [A8].

The calculated midpoint between the R(0) and the P(1) transitions (Table 5) has a value of 2,143 cm^–1. It corresponds with the CO Raman frequency of 2,143 cm^–1 [A9].

In summary, in the rotation-vibration absorption spectra of both heteronuclear diatomic molecules the v = 0 → v´= 1 vibrational transition cannot be detected, that is, the vibration is non IR active. However, the expected frequency of the “missing vibrational absorption band” (estimated by the midpoint between the R(0) and the P(1) transitions) can be observed in the Raman spectra.

So, back to the roots of the theory: To calculate the intensity of vibrational transitions one has to evaluate the vibrational transition dipole momentM_v-v´ for the vibrational transition v → v´

where μ(R) is the molecular dipole moment operator, χ_v (R) and χ_v´ (R) are the nuclear vibrational wave-functions in the vibrational states v and v´ and R are the nuclear coordinates.

The molecular dipole moment is a function of R and can be expanded using a Taylor series expansion about the equilibrium nuclear coordinatesR_e of the initial vibrational state,

where μ(R_e) is the permanent electric dipole moment. With it the vibrational transition dipole moment is given as

Because of the orthogonality of the vibrational wave-functions for v = v´ the integral in the first term is 1, resulting in the permanent electric dipole moment of the molecule μ(R_e) in the vibrational state v.

The function of R is of ungerade (u) [odd] parity, whereas the nuclear vibrational wave-function of the vibrational ground state χ_v(R) is of gerade (g) [even] parity. Hence the second integral will be ≠ 0 only, if the parity of χ_v´(R) is u. In general, the parity of both vibrational wave-functions must be different that the integral will be ≠ 0. In the case v = v´ the parity of both vibrational wave-functions is either g or u and the second integral is zero.

For v ≠ v´ the integral in the first term is zero, because of the orthogonality of the vibrational wave-functions resulting in eq. (25).

With the harmonic oscillator functions the parity selection ruleparity selection rule"?> Δv = ± 1, ± 3, … derives from the second integral.

Therefore, the intensity of a parity allowed vibrational transition is expressed through the first partial derivation of μ(R) and the correct selection rule for a normal vibration to be IR active reads as follows: “During a normal vibration IR radiation can be absorbed only if the condition for the first partial derivation of the molecular electric dipole moment μ(R) is fulfilled.”

To calculate M_v-v´ an accurate electric dipole moment function of symmetric valence bond vibrations in the ground electronic state is required. There are some quantum chemical program packages, but one cannot trust all their results, especially for the v = 0 → v´= 1 transitions.

A possible reason may be a wrong choice of the used electric dipole moment function. As an example of a proofed function for the description of diatomic vibration is the semi-empirical exponential Mecke-type function [A10],

where n and α are empirical parameters to fit the available experimental data best.

Already μ(R_e) = 0 for homonuclear diatomic molecules leads to μ(R) = 0. However, what happens for μ(R_e) > 0?

The exponential Mecke-type function generally exhibits asymptotic behavior as R → 0 and R → ∞. Most important, the function reaches its maximum at R = R_e, that is,

With this consideration the theory supports for all diatomic molecules, that the symmetric valence vibration is non IR active.

In summary, for the correlation with experimental vibration frequencies the key question was, is the symmetric valence vibration in diatomic molecules IR and/or Raman active? As shown by experiment the symmetric valence vibration in diatomic molecules is non IR active irrespective of whether they possess a homonuclear or heteronuclear molecular structure. The exponential Mecke-type function – a nearly accurate electric dipole moment function for the vibration in diatomic molecules – reaches its maximum at R_e and therefore the first partial derivation of μ(R) at R = R_e is zero. This result supports, the symmetric valence vibration in diatomic molecule can be only Raman active and non IR active.