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AbstractsWave functions which are a linear combination of HZ-type elliptical orbitals are optimized to provide either an upper bound or a lower bound to the H$ ground state. For the latter, Temple's formula is used. Three criteria are considered to determine the relative accuracy of these wave functions: ( i ) energy (calculated versus exact eigenvalue) ; (ii) average error; and (iii) local energy. Although the lower-bound optimized wave functions obtained are the most accurate available for H$ from approximate wave functions, they are still inferior to the corresponding upper-bound wave functions by criteria (i) and (ii). I n particular, using criterion (ii), it is shown numerically that the upper-bound functions are "correct to second order," while the lower-bound functions are almost, but not quite, "correct to second order." Despite this, the local energy analysis, criterion (iii), reveals that the lower-bound wave functions can be more accurate than the upper-bound functions in some regions of space, and hence give more accurate values for physical properties sensitive to these regions. Examples considered are the dipole-dipole and Fermi contact interactions. Des fonctions d'ondes

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