A m e th o d is described for c alc u latin g rigorous low er b o u n d s to p o larizab ilities a n d o th e r second-order p ro p ertie s ev en w hen, as u su al, th e field-free (u n p e rtu rb e d ) p ro b lem c a n n o t be solved ex actly . T his m e th o d h a s b een a p p lie d to calc u la te dipole p o larizab ilities fo r th e low -lying singlet a n d trip le t S sta te s o f th e tw o -electro n ions H " , H e, a n d L i+, u sin g wellc o rrelated v a ria tio n a l tr ia l fu n

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... n s u p to 104 te rm s in len g th . A s a b y p ro d u c t o f th e rigorous b o u n d s we o b ta in new h ig h -accu racy estim a te s, b a se d on th e c o n v en tio n al v a ria tio n -p e rtu rb a tio n procedure, w hich larg ely s u p p la n t p rev io u s c alc u latio n s on th e se ions. T h e low er b o u n d s a re a c c u ra te to w ith in a sm all fra c tio n o f a p e r c e n t for all species e x c e p t H~ a n d th e m e ta sta b le 2X S s ta te o f H e (w here th e a c c u ra c y is a b o u t 2 a n d 1 % resp ectiv ely ), a n d th is accu racy is sufficient to ru le o u t m a n y p rev io u s th e o re tic a l a n d e x p e rim e n ta l resu lts. T h e calc u lated values are th e n em ployed to e v a lu a te th e lead in g u n k n o w n te rm s in th e Z _1 expan sio n o f th e p o larizab ility , th e re b y e x te n d in g th e h ig h -a c c u ra c y tr e a tm e n t to all m em bers of th e isoelectronic sequence.