General 1e Derivative Property Analysis (DPROP)
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Reference
E. D. Glendening and F. Weinhold, J. Comp. Chem. ??, ???? (2010?).
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Notes
The DPROP (derivative property) keyword provides a powerful and versatile tool to perform NBO analysis of nth-order derivative properties for any chosen n (within numerical accuracy limits) and 1e property that is interactively available from the host ESS program or input .gen file (cf. PROP keyword for allowed properties). For the x-component of the electric dipole operator (PROP = DIPOLE), for example, the corresponding DPROP evaluations correspond to polarizability (αxx) for n = 1, hyperpolarizability (βxxx) for n = 2, and so forth.Derivative properties are evaluated by finite-field perturbation techniques, analogous to evaluation of dipole moment (P = DIPOLE operator) as the limiting energetic response to a finite electric field of strength λ, viz.,
H(λ) = H(0) + λP
<P> = d<H(λ)>/dλ ≈ [<H(λ)> - <H(0)>]/λ as λ → 0 where H(0) is the field-free Hamiltonian operator and λ is chosen sufficiently small for reliable numerical differencing. For higher-order derivative properties, one can similarly evaluate <H(-λ)>, <H(2λ)>, <H(-2λ)>... to obtain corresponding estimates for nth-order derivatives by standard numerical differencing formulas. However, as n increases, the choice of "sufficiently small" λ becomes increasingly problematic, due to accumulating round-off errors in sequences of finite-length arithmetic differencing operations. Selecting a suitable λ value for the chosen operator P and order n usually requires prudent trial and error to achieve satisfactory numerical reliability, but these numerical considerations are independent of how DPROP results are presented in the NBO output for user-selected n, λ values.
DPROP derivative evaluation and analysis of a given property for chosen n and λ is requested by keyword and bracket-list input of the form
DPROP=property <n/λ> If the bracket-list is omitted, default values n = 1, λ = 0.01 a.u. are assumed.
DPROP output resembles PROP output in expressing the overall property as a sum of Lewis (L) and non-Lewis (NL) contributions, with subsidiary correlation correction for post-SCF methods. The property is thereby described as a sum of bond-properties (NBO contributions), each modified by delocalization (NLMO increments or decrements) and electron-correlation corrections to give the final numerical value.
DPROP keyword output for the electric polarizability (DPROP=DIPOLE <1/0.01>) is illustrated below in sample output for the formamide molecule (RHF/3-21G level). See the NBO 6.0 Manual, pp. B???? for additional discussion of DPROP (and related PROP) keyword usage.
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Sample Gaussian Input
#rhf/3-21g pop=nboread
RHF/3-21G for formamide (H2NCHO)
0 1
H -1.91338 -0.34957 0.00000
H -0.65570 -1.51410 0.00000
N -0.94020 -0.55750 0.00000
C 0.00000 0.41595 0.00000
O 1.19830 0.23282 0.00000
H -0.43592 1.40790 0.00000
$nbo dprop=dipole <1/0.01> $end