Domain: Computational chemistry

A solution-phase \\(SN_2\\) reaction in acetonitrile at 298 K is studied with the protocol: optimize and do thermochemistry at\\(\omega\mathrm{B97X-D/def2-TZVP/CPCM(ACN)}\\); single-point energies at \\(\mathrm{DLPNO-CCSD(T)/def2-TZVPP/CPCM(ACN)}\\). Quasi-harmonic entropy with a \\(100\ \mathrm{cm^{-1}}\\) floor is used; ZPE is scaled by \\(0.985\\) (apply to all species); D4 dispersion is already included in the \\(\omega\mathrm{B97X-D}\\) energies (do not add it again); DLPNO single-points replace only the electronic energy (do not recompute thermochemistry at DLPNO). The transition state has exactly one imaginary frequency \\((256i\ \mathrm{cm^{-1}})\\). The 298 K Gibbs free energies at 1 atm reference constructed from the above(i.e., \\(G^{1\ \mathrm{atm}}_{298} = E^{\text{DLPNO}}_{\text{elec}} + G^{\omega\mathrm{B97X-D}}_{\text{thermo}}\\)) are* \\(G_{298}^{1\ \mathrm{atm}}(\mathrm{CH_3I}) = -250.150000\ \text{hartree}\\)* \\(G_{298}^{1\ \mathrm{atm}}(\mathrm{Cl^-}) = -63.460000\ \text{hartree}\\)* \\(G_{298}^{1\ \mathrm{atm}}(\mathrm{TS}) = -313.557000\ \text{hartree}\\)Use the correct standard-state change for a bimolecular reactant state going to a unimolecular TS: \\(\Delta n = 1-2 = -1\\) and at 298 K, \\(RT\ln(24.46)=1.89\ \mathrm{kcal,mol^{-1}}\\). Report \\(\Delta G^{\ddagger}_{298}\\) at 1 M standard state in \\(\mathrm{kcal,mol^{-1}}\\).

Grading rubric:
criterion: Correctly identifies the translational partition function as the specific thermodynamic source of the standard state correction term.
weight: 20
criterion: Correctly validates that empirical dispersion corrections are excluded from the DLPNO-CCSD(T) term to avoid double-counting electron correlation.
weight: 20
criterion: Correctly defines the quasi-harmonic approximation as a remedy for the asymptotic singularity of the harmonic entropy function at low frequencies.
weight: 10
criterion: Correctly applies the vibrational scaling factor of 0.985 specifically to the zero-point energy component of the thermodynamic correction.
weight: 20
criterion: Correctly characterizes the transition state topology as a first-order saddle point possessing exactly one imaginary vibrational mode.
weight: 10
criterion: Correctly computes the molar volume correction term of 1.89 kcal mol^-1 derived directly from the ideal gas law isotherm at 298 K.
weight: 10
criterion: Correctly reports the final activation free energy value strictly within the range of 31.35 to 31.40 kcal mol^-1.
weight: 10

Answer:
31.37 kcal mol^-1