First derivatives of the energy generalized forces

This follows because of the variational condition (3). Although the wave function p changes with X, the effect of that change upon the energy is zero to first order. The constant X may be a true constant, such as the electron mass or charge, or it may be a quantity which is held fixed in the determination of the ground state but which can be varied physically, such as the position of an atom. Each type of variation has useful consequences. 

The well-known Heilmann-Feynman theorem for the force on a nucleus results from choosing X to be the position of a nucleus R,. If 4 is the exact wave function calculated holding R, fixed, then the force on nucleus I is 

Other types of variations lead to useful expressions as well. For 

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