10.9 Harmonic Vibrational Analysis

10.9.3 Resonance-Raman intensities

The theory of resonance-Raman spectroscopy is fully described by the Kramers-Heisenberg-Dirac dispersion formalism based on the Raman polarizability tensorKelley:2008

αστ(ωL,ωS)=υ[f|r^σ|υυ|r^τ|iωL-ωυi+iΓiυ+f|r^τ|υυ|r^σ|iωυi+ωS+iΓiυ] (10.19)

between initial state |i and final state |f. Here, ωL and ωS are the frequencies of the laser (incident photon) and of the scattered photon, respectively. Eq. (10.19) is inconvenient due to the sum over intermediate states υ (vibrational levels on all accessible electronic states), and the usual procedure is to expand the static molecular polarizability as a Taylor series in the normal coordinates,Albrecht:1961, Tang:1970 which allows the Raman intensity to be decomposed into Franck-Condon (or “A-term”) contributions and coordinate-dependent Herzberg-Teller (“B”- and “C”-term) contributions.Albrecht:1961, Tang:1970, Kelley:2008 Nevertheless, each term contains sums over intermediate vibrational states and becomes difficult to evaluate for large molecules with numerous vibrational modes.

As such, in most cases only the lowest-lying Raman-active excited electronic state is considered in computing the RR spectrum. In principle one should consider the effects of Duschinsky rotation,Sharp:1964 i.e., the fact that the normal modes are different in each electronic state. Neglecting this effect for simplicity and thus using ground-state normal modes only, one arrives at the “independent-mode, displaced harmonic oscillator” (IMDHO) model,Petrenko:2007 in which resonant enhancements to the vibrational intensities (for modes 1 and 2, say) are expressed as ratiosHeller:1982, Myers:1996, Dasgupta:2019

I1I2(ω1gΔ1ω2gΔ2)2. (10.20)

In this equation, ω1g and ω2g represent the ground-state vibrational frequencies for normal modes Q1 and Q2 and ω is the electronic excitation energy. The first equality in Eq. (10.20), written as an approximation here, is exact within the IMDHO model. The quantity

Δk=(ωk)1/2ΔQk (10.21)

evaluated at the ground-state geometry (𝐐=𝟎), is the slope of the excited-state potential energy surface along mode k. This leads to the second equality in Eq. (10.20).

The time-dependent picture provides means to derive this expression.Heller:1982 In this approach, the requisite polarizability tensor elements involving different electronic states are expressed as the Fourier transformation of the time-evolving overlap between initial- and final-state electronic wave functions:

α(ωL)0eiωLt-Γtψf|ψi(t)𝑑t+NRT. (10.22)

Here, “NRT” indicates the non-resonant terms that are neglected in RR spectroscopy. Large molecules likely spend no more than 10–20 fs in the Franck-Condon region and the overlap integral is likely only significant on that timescale.Heller:1982, Myers:1996, Dasgupta:2019 Within a model that considers only two electronic states, the RR intensity that one obtains is

IkωL(ωL-ωk)3(ωkΔk)2 (10.23)

where μk is the reduced mass of the kth normal mode.

Assuming identical force constants for Qk in both the ground and excited electronic state, one obtains a linear transformation between the displacement Δk of the equilibrium position of this mode, expressed in normal coordinates, and the displacements Δ~i expressed in Cartesian coordinates:

Δ~i=k=13N-6(Likmi1/2)Δk. (10.24)

In matrix form this is

𝚫Q=λ-1𝐋𝐌1/2𝐕X (10.25)

where λ is the eigenvalues of mass-weighted Hessian matrix, M defines the matrix of atomic masses and VX is the energy gradient in Cartesian coordinate Raman intensities are related to the dimensionless displacements

Δk=(λkme)1/4ΔQk. (10.26)

JOBTYPE set to RRAMAN invokes the calculation of resonance-Raman intensities.

       Controls whether frequency job calculates resonance-Raman intensities
       False Normalise RR intensities True Doesn’t normalise RR intensities

Example 10.18  Calculating resonance-Raman intensities.

0 1
  C         1.8288506578   -0.1219336002    0.0000000000
  C         0.6155951063    0.3987918905    0.0000000000
  C        -0.6155955606   -0.3987931260    0.0000000000
  C        -1.8288502653    0.1219348794    0.0000000000
  H         2.7085214046    0.4909328271    0.0000000000
  H         1.9881851899   -1.1843222290    0.0000000000
  H         0.4885913610    1.4671254626    0.0000000000
  H        -0.4885933454   -1.4671268234    0.0000000000
  H        -1.9881816088    1.1843239478    0.0000000000
  H        -2.7085226822   -0.4909289672    0.0000000000

 method          hf
 jobtype         RRAMAN !Jobtype resonance Raman
 basis           3-21G
 cis_n_roots 1
 cis_state_derivative 1 ! Excited state