First-Order Geometrical Response Equations for State ... - Jack Simons

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374 BAK, BOATZ, AND SIMONSThis matrix is easily shown to be unitary:and the larger matrix defined as= -U+U = l , (55)lmol ~ OU= O U O ,( O O lin)(56)with lmoland lin being identity matrices with the dimensions of All and A33,respectively,and,therefore, algo unitary. Note thatthis unitary matrix bas earlierbeen used by Lengsfield [5] in deriving second-order MCSCF theory for SAwavefunctions. Using the matrix' to perform a unitary transformation on Eq. (52)gives'-'-""'"_.- - - - -,UAU+UV= uii.. (57)'Oefining--- -A ==UAU+,V ==UV,B ==UB,(58a)(58b)(58c). Eq. (57) readsAV = B. (59)The A matrix and the V and B vectors are naturally written in subblocks:,All A21+A= A21 A22( A31 OA31+O ,A33)(60a), V =(~:),(60b)B=(::).(60c)where the elements of the blocks are easily evaluated. Using the followingtwadefinitions,fiLII == 2:Cfcf,(6ta)RMIl ==8u - Lu,(6tb)
FIRST-ORDER GEOMETRICAL RESPONSE EQUATIONS 375which algo result in the following equation:MIJA= Lcfcf,pthe final working expressions for the elements obtained areA u - Y SA Y SA Y SA + Y SA + ~ SA ~ SA ~ SA + ~ SAij. nk - ijnk - jink - ijkn jikn UikSnj - UjkSni -UinSkj UjnSki ,(63a)(62)Ajk,nk = WRL Md.T~f- Tl~),"I .Ajk.sL = WRSSR( ~MKIMJLHIJ - MKLER + ZLKL) ,A.~~.nk= (WR - Ws) L cJ(T~f - Tl~),JA~hu = SRTSSU(WR - ws) (Es - ER),- .(63b)(63c)(63d)(63e)Bi} = L L S~HYi~n - Yj~n + SinSkf- SjnSmn k
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First-Order Geometrical Response Equations for State ... - Jack Simons

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