Analytic derivative theory based on the Hellmann–Feynman theorem

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Abstract
General formulae for the second, third, and fourth derivatives of the energy with respect to the nuclear coordinates of a molecule are derived from the Hellmann–Feynman theorem. Hurley's condition is used to obtain approximations to the first-order wavefunction, from which the second, third, and fourth energies can be obtained, leading to quadratic, cubic, and quartic force constants. The procedure is equivalent to minimizing the derivative energy by perturbed variation techniques. The expressions for these higher energy derivatives are much simpler than those of the direct analytic derivative method. The electrostatic calculation involves only one-electron integrals. The coupled Hartree–Fock equations to obtain the wavefunction derivatives become much simpler. The present theory provides a great conceptual simplification. However, the theory is correct only if the basis set is complete or basis functions are independent of the perturbation. Keywords: analytic derivative theory, Hellmann–Feynman theorem, force constants, the curvature theorem.

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