A Deduction of the Hellmann-Feynman Theorem

In this paper we present a deduction of the Hellmann-Feynman (HF) theorem for the lowest eigenenergy E0λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{0}\left (\lambda \right ) $\end{document} of a Hamiltonian Hλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ H\left (\lambda \right ) $\end{document}, that is : its second-order derivative with respect to he parameter λ,∂2E0∂λ2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda ,\frac {\partial ^{2}E_{0}}{\partial \lambda ^{2}},$\end{document} is always less than the expectation value of ∂2Hλ∂λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {\partial ^{2}H\left (\lambda \right ) }{\partial \lambda ^{2}}$\end{document} in the ground state. We also point out that the above deduction does not hold for the FH theorem in ensemble average. The electric polarizability of molecules is studied by the deduction of the HF theorem