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

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366 BAK, BOATZ, AND SIMONSUsing Eq. (1)and the orthonormality of the MO'Sgives the followingidentity:oKiaKjO = ~ 7(Xl'lepj) + ~ 7(ep;jXv)I' ua v ua+ Si},(18)where we have definedSij ==~Ki I' Kja(XI'IXv) vI'Voa'(19)The term.caD, of course, be expressed as alineat~ aK~X~ aa I'aKi~I' oa1'.]combination of MO'S. ~ -2X = ~ Ulepi>(20)which, for realorbitais,definesU; = ~ a;(Xl'lepj).Equation (18) caD then be compactly expressed as0= Ui} + U; + Si}'Using this notation, we caD rewrite (ahiN(aa) and rafij IkI)]/(aa) as(21)(22) -andwhereahij- ha ~(u a h U a h ),,- ij+~ kikj+ kjik,uaka(ij IkI) = (ij IkI)a + ~ (U::li(mj IkI) + U::lj(im IkI)aam+ U::lkW ImI) + U::.lWIkm»,h!'. I] == ~ K I' i K jahl'V -I'V v oa '(23a)(23b)(24a)(ij IkW == ~ K~ KtK; K~ a(JLV Ipu ).~ aaSimilady, using Eq. (2) and the orthonormality of statesgivesO- ~ ac1 CB ~C A oC!- ~- l + ~ l-.l oa l oa(24b)(25)
FIRST-ORDER GEOMETRICAL RESPONSE EQUATIONS 367Since a change of a CIvector musi be expressible in the space of all CIvectors, wehaveA N DacI = L VADCI ,aa D(26)which can be inserted in Eqo (25) to giveO = VAB + VBA. (27)Since the one- and two-particle coupling coefficients both ale independent ofnuclear geometry (as long as the nuclear framework. belongs to the point groupchosen for the calculation), using Eqs. (17c) and (26), the density matrices at thedisplaced geometry ale given to. first order in A asNl'(x + A)fJ= l'ff+ AL VRAL cfc1(l'b' + l'GI)+ ..., (28a)A IJNr(x + A)ffkl= rffkl + AL VRAL cfc1(rfftcl + rGil) + ....A II(28b)Notice that if one wishes to include distortions that reduce the symmetry of thenuclear framework, one bas to evaluate the coupling coefficients in the lowersymmetry group. In general, if all possible distortions ale required, the calcula~tion bas to be carried out without use of point-group symmetry.Now that we have evaluated how different quantities at an infinitesimally displacedgeometry ale expressed to first order in A, and have shown in Eqs. (22)and (27) how the orthonormality conditions of the orbitais and states reduce thenumberof independent variabies, we ale ready to derive the response equationsfor the-set of SAwave functions. To obtain these equations, ope bas to take advantageof the two conditions that must be fulfilled for converged SA-MCSCF wavefunctions at aU nuclear geometries:, ,. (A) the SA-GliTcondition expressed inEq. (16),that states that the orbitalsareoptimized in a weighted SAmanner, and(B) the CIcoefficients for each of the SA-MCSCF internat states have to be variationallyoptimized in the chosen space of CSF'S;that is,l)RBER= L cfc! BIJ,IIbas to be valid for all CIcoefficients of the internat states.(29)In the next section, we derive the equations that arise from the GBTCondition;the equations thatarise from Eq. (29) wIll be evaluated in Section VI.V. First-Order Responses Arising erom the GBTImposing the SA-GBTto be fulfilled at the infinitesimally displaced geometry,we can use Eqs. (lO), (14), (15), (17d), (17e), (23), and (28) in Eq. (16) to giveus this GBTcondition in powers of A. Collecting first-order tetms generates the
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First-Order Geometrical Response Equations for State ... - Jack Simons

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