μ/F, α and β of PA vs a finite field F (in a.u.) applied along the non-periodic transverse (y; left) and perpendicular (z; right) directions. Calculations at the field-free optimized structure provide the electronic contributions to α and γ (α e and γ e , respectively) as coefficients in the expansions (2), (7), and (9). Structural relaxation in the presence of the field (R F ) provides, in addition, the vibrational contributions according to Eqs. (3), (8), and (10). Calculations performed at the HF/[6-31G+B] level. Computational parameters: T E = 10, T C = 10, T x = 100, S = 30.
Independent components of PA nuclear relaxation second hyperpolarizability tensor, . There are four classes (first column), depending upon the number of repeated indices and three subgroups within each class. The latter depend upon whether the static field(s) in the nonlinear process are associated with 4 (column (a)), 2 (column (b)), or 1 (column(c)) directional indices (in capital letters). Components that vanish due to planar symmetry are included in parentheses. Of the remainder only the highlighted ones were calculated as they do not require application of a finite field along the periodic direction in the FF-NR method. The row labelled specifies the multiplicity. There are N ind independent components, of which do not vanish due to symmetry, and N calc were computed in the present work.
Static electronic (hyper)polarizability tensors, α e and γ e (in a.u.), of PA calculated by the CPHF method at different basis set levels. The 6-31G basis set has been used as a starting point and added with ghost functions progressively increasing in number and arranged according different (y, z) configurations with the origin at the mid-point of the carbon−carbon double bond and z perpendicular to the molecular plane. Configuration A includes 4 ghost molecules/cell at (0.0, ±1.5) and (± 3.5, 0.0) Å ; configuration B includes 10 ghost molecules/cell at (0.0, ±1.5), (± 3.5, 0.0), (0.0, ±3.0), and (± 2.0, ±3.5) Å ; configuration C includes 12 ghost molecules/cell at (0.0, ±1.5), (± 3.5, 0.0), (0.0, ±3.0), (± 2.0, ±3.5), and (± 7.0, 0.0) Å . Column |Δ| reports the percentage improvement between the numbers on the leftside column and those at the previous step. Values of the second hyperpolarizability along the longitudinal direction x in 106 a.u. Computational parameters: T E = 10, T C = 10, T x = 100, S = 30.
Static electronic CPHF (hyper)polarizabilities – α e and γ e (in a.u.) – of ethylene calculated by using different molecular basis sets, specifically optimized for the estimation of optical properties.36–39 Comparison with the split-valence 6-31G basis set and the [6-31G+B] set (used for periodic 1D calculations) is made. Computational parameters as in Table II.
Static electronic polarizability α e (in a.u.) of PA calculated by the CPHF and FF methods at the field-free equilibrium geometry. Calculations have been performed using the [6-31G+B] basis set. Equation (2) was employed to fit the dipole moment versus the finite field to provide numerical estimates of α e . Computational parameters as in Table II.
Static electronic second hyperpolarizability γ e (in a.u.) of PA calculated at the field-free equilibrium geometry by the CPHF and FF methods. Columns (a), (b), and (c) report numerical estimates obtained by fitting the dipole moment, polarizability, and first hyperpolarizability (Eqs. (2), (7), and (9)), respectively, versus a finite field. The four classes of components (i. - iv.) are defined in Section II C and Table I. Basis set and computational details as in Table IV.
Static vibrational (nuclear relaxation), α nr , and electronic, α e contributions to the independent, non-vanishing polarizability components of PA. Analytical values obtained using the Berry phase approach (α nr ) and the CPHF method (α e ) are reported. Basis set and computational details as in Table IV.
FF-NR static and dynamic vibrational (nuclear relaxation) contributions to the second hyperpolarizability γ (in a.u.) of PA obtained by fitting (a) the dipole moment, (b) the polarizability and (c) the first hyperpolarizability versus the finite field according to Eqs. (3), (8), and (9). The uncertainty in the fitted parameters is, generally, within 5%; dashes indicate components that are currently inaccessible. For basis set, computational, and other details see Table IV.
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