14.3 Calculation of VCD Spectra

VCD intensities are proportional to the rotational strengths, which are defined as scalar product between the electric and magnetic transition dipole moments. The rotational strength for the transition of the nth vibrational normal mode in the electronic ground state is

R  = Im (μ    ⋅μ     ).
  n       el,n   mag,n

Whitin the harmonic approximation, the electric and magnetic transition dipole moments can be written as

           ∘ ---
  (μ   ) =   -ℏ-∑  P λS λ
    el,n β    ωn  λα  αβ  α,n
             ∘ -----∑    λ  λ
(μmag,n)β = -  2ℏ3ωn    M αβSα,n.

Pαβλ is the so called atomic polar tensor (APT), Mαβλ the so called atomic axial tensor (AAT) and Sα,nλ the transformation matrix from cartesian coordinates to normal coordinates.
ωn is the frequency of the nth vibrational normal mode, α and β describe cartesian coordinates and λ labels the atom nuclei.

Both the APT and the AAT are devided in an electronic and a nuclear contribution. According to Chem. Phys. Lett. 252, 221 (1996), the electronic contribution of the AAT within the coupled perturbed Hartree-Fock (CPHF) formalism is given by

      N∑occ N∑bf [               ∑N
Iλαβ =          cμicνi⟨χRμλα|χBνβ⟩+    cμicνpU Bipβ⟨χRμλα|χν⟩
      i=1μ,ν=1                 p=1
                 ∑N                    ∑N                     ]
               +    cμpcνiURipλα⟨χ μ|χBνβ⟩ +     cμpcνqURipλαUBiqβ⟨χμ|χν⟩ .
                 p=1                   p,q=1

To compute VCD spectra with TURBOMOLE, mpshift and subsequently aoforce have to be called. First, if $vcd is set, mpshift calculates the disturbed UipB. In the second step vibrational frequencies, normal modes and VCD rotational strengths are calculated by aoforce. At present, calculation of VCD spectra is possible at Hartree-Fock and DFT level (for LDA, GGA and hybrid-GGA functionals) in C1 symmetry. Since the angle between the electric and magnetic transition dipole moments is not gauche invariant, it will only be printed out if the distance between the center of mass and the origin is smaller than 1e-03 a0. The script VCD can be used to perform these two steps automatically.