Regional stationary principles and regional virial theorems

If the Born‐Oppenheimer Hamiltonian operator for a molecular system is averaged over all space for electrons $\text{2,3, ···\hspace{0.17em}, N}$ , but for electron 1 is averaged only over the volume A, with surface S, it is shown that the regional expectation value so defined is stationary with respect to change of a parameter $\zeta$ in the exact wavefunction, provided that ${\int }_S{\mathrm{[(\partial /\partial \; n}}_1{\mathrm{)\hspace{0.17em}P}}_{\zeta }{\text{(1,1′)]}}_{\text{1=1′}}{\mathrm{\hspace{0.17em}dS}}_1\text{=0}$ , where, for nodeless $\psi$ , $$P_{\zeta }\text{(1,1′)=∫ ··· ∫ψ*(1′,2,···\hspace{0.17em}, N) ψ (1,2, ···\hspace{0.17em},\hspace{0.17em}N) ×}\frac{\mathrm{\partial \hspace{0.17em}}\ln \text{[ψ(1, 2, ···\hspace{0.17em}, N)/ ψ*(1′, 2, ···\hspace{0.17em}, N)]}}{\mathrm{\partial \; \zeta }}{|}_{\text{ζ=1}}{\mathrm{d\tau }}_2{\mathrm{···\; d\tau }}_N$$ . Under this condition with $\zeta$ a scale parameter, a regional virial theorem is shown to hold for the volume A, in the sense previously described by Bader and co‐workers [J. Am. Chem. Soc. 93, 3095 (1971): Chem. Phys. Lett. 8, 29 (1971), J. Chem. Phys. 56, 3320 (1972); 58, 557 (1973)]. This condition differs from the corresponding condition suggested by Bader ${\mathrm{(\partial \; /\; \partial \; n}}_1\mathrm{)\rho \; (}{\mathbf{r}}_1\text{)=0}$ on S, but in most cases the surfaces determined by it should be similar to those determined by Bader's condition. Corresponding conditions for approximate wavefunctions are discussed, and illustrations are given.