Extension of the Karplus Relationship for NMR Spin−Spin Coupling Constants to Nonplanar Ring Systems: Pseudorotation of Cyclopentane

A new Karplus equation for pseudorotating five-membered rings is derived by expanding calculated NMR spin−spin coupling constants (SSCCs) J as a function J ( q,φ) of the puckering amplitude q and the pseudorotational phase angle φ. The approach was tested for cyclopentane but is equally applicable to ribose sugars and other biochemically interesting five-membered rings. It is based on the calculation of the conformational potential V ( q,φ), which in the case of cyclopentane was determined at the MBPT(2)/cc-pVTZ level of theory. Cyclopentane is a free pseudorotor (barriers Δ E and Δ H ≤ 0.01 kcal/mol) with a puckering amplitude q = 0.43 Å and a barrier to inversion Δ H = 5.1 kcal/mol, in perfect agreement with experimental data. The SSCCs of cyclopentane were calculated at MBPT(2)/cc-pVTZ geometries by use of coupled perturbed density functional theory (CP-DFT) with the B3LYP functional and a (9s,5p,1d/5s,1p)[6s,4p,1d/3s,1p] basis set. In addition, coupled-cluster singles and doubles (CCSD) calculations were carried out to verify the CP-DFT results. All geometrical parameters and the 10 SSCCs of cyclopentane are determined as functions of the phase angle φ and averaged to give 〈 n J 〉 values that can be compared with experimental data. The following SSCCs (in hertz) were obtained at CP-DFT/B3LYP/[6s,4p,1d/3s,1p]: 〈 1 J (CC)〉 = 34.0; 〈 1 J (CH)〉 = 127.6, exp 128.2; 〈 2 J (CCC)〉 = 2.3, exp (+)2.8; 〈 2 J (CCH)〉 = −2.6, exp (−)3.0; 〈 2 J (HCH)〉 = −12.4, exp (−)12.4; 〈 3 J (CCCH)〉 = 3.9; 〈 3 J (HCCH, cis )〉 = 7.7, exp 7.7; 〈 3 J (HCCH, trans )〉 = 5.6, exp 6.3; 〈 4 J (HCCCH, cis )〉 = 0.1; 〈 4 J (HCCCH, trans )〉 = −0.6. Magnitude and trends in calculated SSCCs are dominated by the Fermi contact term [with the exception of 1 J (CC)]. A new way of determining puckering amplitude and pseudorotational angle by a combination of measured and calculated SSCCs is presented.