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 tensor467

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,15, 947 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.15, 947, 467 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,876 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,747 in which resonant enhancements to the vibrational intensities (for modes 1 and 2, say) are expressed as ratios368, 675, 204

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.368 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.368, 675, 204 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 10.19  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