- Z , <0°.|H|0°> % (2.55) t€S t,s€S A where 0?, 0°, in these summations run over eigenvectors in the x s space S A only. On transforming these vectors to a new set: (r = 1, ..., n A), which diagonalizes H in S A, eq. (2.55) becomes n = i = £ A o 2 . (2.56) n=l n If f is an exact solution of (2.16), uncoupling the parts of the 0°, (t = 1. n A)» in S A, and the 0p, (p « 1, • •», n^) in Sg, exactly, then the ^ i n (2.56) are exact eigenvectors of H> and each o n is: identically zero. If f is not exact, then the ^ w i l l be only approximate eigenvectors of H, and o o is the variance of H with respect to the single approximate 2 eigenvector yn+ Thus o is the * *AA- (2.61) Substitution of this equation into the BA block equation of (2.60) gives f i n terms of f and V only, *' = \" 1 \" VAA * ^ A B ^ l A + ' and thus, to f i r s t order in f, F \" • VBA VIA + < VBB \" V I A W ^ I A + Thus, i f f is small, the transformation (2.62) is nearly linear, although not homogeneous. 30 2.2 Effective Operators 2.2.a Basic Definitions The primary application of the partitioning formalism just described is in the construction of effective operators. In this context, such operators are defined in either of the sub-spaces S A or Sg. of the f u l l basis space, but their eigenvalues form a subset of the eigenvalues of the original operator i n the f u l l basis space, and the corresponding eigenvectors are related im some way to those of the original operator. There are two ways of regarding the matrices of such effective opera-tors. They can be regarded as the matrix of a transformed operator i n the old basis (active sense), or, alternatively, as the matrix of the old operator in a transformed basis (passive sense). Both points of view are equivalent, but i n , what follows, the former w i l l be emphasized.. The: simplest set of such effective operators for the matrix H has already been defined i n equations (2.14) and (2.15). In S A, we have the operator ftAaHAA+HABf> ( 2 * 6 5 a ) with the eigenvalue equation fi*XAA-XAA ? U ) ' ( 2 ' 6 5 b ) and i n S B» ftB * HBB \" ( 2 * 6 6 a ) with the eigenvalue equation, 31. Both and Hg. are nonfhermitian in general, although their eigenvalues f ^ and f are real, since they are subsets of the eigenvalues of the hermitian operator H* The eigen-vectors X ^ and Xgg, are not orthonormal in general, because they are truncations of the orthonormal eigenvectors X of the f u l l hamiltonian H. It is possible to derive a pair of self-adjoint effective operators directly from the eigenvalue equation (2.1a). Pre-multiplicationi by T + , and use of eq. (2.17), yields, °AXAA * *L*Xk f < A ) ' <2'67a) where and, G A - (T'HT)^ = H a a * K A f if • fFHBA • fFHBB:f, (2.67b) where V B B * % XBB f U ) » < 2' 6 8 a ) G B « (T+HT)Bg. = H M - H B A f t - f H A B • f H ^ f * . (2.68b) * t * The off-diagonal block of T HT Is given by, GBA * «m + V - ™ U \" '\"AB*' f2'6\" which i s just the quantity D(f), defined in; eq. (2.16). When GgA « 0, i t can be shown that, GA \" gA*A» QB. \" g B % » ( 2 * 7 0 ) using eqs. (2.9)» and the definitions of the effective operators presented above* Thus, when f is known exactly, the se l f -adjoint effective operators G A and Gg could be considered to be obtained from the non-selfadjoint effective operators H A 32. and Hg; by orthonormalizing the eigenvectors of the l a t t e r . I t i s also possible to obtain, s e l f - a d j o i n t e f f e c t i v e operators i n S A and Sg. by orthonormalizing the truncated eigenvectors. The e f f e c t i v e operators H A, Hg,,, and G A and Gfi above,, are uniquely determined once p a r t i c u l a r p a r t i t i o n i n g s of the basis and eigenvector spaces are defined. The s e l f -ad'joint e f f e c t i v e operators obtained by orthonormalization are not unique, however, i n that they depend on the p a r t i c u l a r orthonormalizationi procedure employed. The symmetrical orthogonalization procedure of Lowdin (1970) and others, has the feature that the new orthonormalized vectors resemble the i n i t i a l vectors as c l o s e l y as possible, i n a p a r t i c u l a r sense. 1 Applied to the present case, the new orthonormal eigenvectors are given by g * X.» , (2.71a) \"AA \" &A \"AA i n S A, and, CBB = «B* XBB • ( 2 ' 7 1 b ) i n Sg. Thus one has, CAA CAA * XAA gA XAA = 1k * (2.72) CBB CBB = XBB gB XBB = 1B • by eq* (2.8).. The eigenvalue equation i n i s obtained either by premultiplying (2.65b) by g A or (2.67a) by g A to get, * I n the notation used above, the difference between the two sets i s measured by ^ i j \" X i j ' which i s minimized i f G i s given by eq. (2.71) (Lowdin, 1970).. 33. SA CAA = CAA (A) where ~ 4 * -4 H A - S A * W (2.73) (2.74a) (2.74b) Similarly, premultiplicatiorri of (2.66b) by gg or (2.68a) by -4 gg . gives the equation (B) HB CBB * CBB f (2.75) where HB - gg* ifggg* - 4 - 4 (2.76a) (2.76b) It is also possible to define effective operators in, either S A or Sg for any other operator defined i n the unpartl-tioned space. For some operator M, (2.77) where I 1 4 V ] MAA MAB MBA MBB f = MAA + MAB f + f MBA + f MBB f' (2.78) MA has the same form in M as 0 A defined in (2.67b) has in H. Here M*A has the same expectation values for the truncated eigenvectors X A A as the operator M has for the f u l l eigen-vectors X. . An effective operator with the same properties with respect to the orthonormalized eigenvectors is clearly given by, 34. which i s analogous to R\"A defined im eq. (2.74).- The analogue of the e f f e c t i v e operator H A of eq. (2.65a) can be obtained by premultiplying MA by g * , following eq. (2.67b). E f f e c t i v e operators f o r M r e s t r i c t e d to Sg, analogous to (2.77) - (2.79), can be obtained i n a s i m i l a r manner. 2.2.b: Eigenvectors and Eigenvalues of the E f f e c t i v e Operators In order to amplify the material i n the immediately pre-ceding subsection, the connection between the eigenvalues and eigenvectors of the operator H and those of the e f f e c t i v e operators H, G, and H, w i l l be i l l u s t r a t e d here from a d i f f -erent point of view. The f u l l operator H has the eigenvalue equation 1 T l 1 (2.80) Once the two basis spaces, S A and Sg, are defined, each eigen-vector can be written as a sum of two parts, t i - t i A * f i B . (2.81) one part i n S A and one i n Sg*, The eigenvectors are them-selves divided into two sets, 1 P J ^ * ( i = 1, •-••»». n A ) , and » ( i s 1* *••» n B ) , where n A + n f i = n, according as they l i e i n S A or Sg. The basic property of f i s to map the part of ' f ' ^ * n SA * n i t o t h e p a r x * n SB» a c c o r d i n S x 0 35. V[B S f f±K* (2.82a) S i m i l a r l y , one has,. - ( - f * ) ^ ? * (2.82b) Combination with eq.. (2.81) y i e l d s , ^ U ) . ^(A) . ^(A) . ^ ( A ) + f ^ ( A > - ( 1 A * t)V{£h (2.83a) and, * ( l f i - f f ) f i B - (2.83b) Ini the notation! used here and throughbut t h i s subsection, the operator f i s to be regarded, when necessary, as embedded i n the n-dimensional basis space, but w i l l , be denoted by the same symbol as before. The eigenvalue equation; f o r H A i s ftA ^ i f * J i ^ i A ^ ( i \" nA>* ( 2 ' 8 * a ) where the eigenvectors s a t i s f y < ^ i A ) l g A l ^ J A } > \" 6iF U , j B l r n A K t'2.8iH») For the e f f e c t i v e operator G A, i t i s eA ^ U ) ' Ii*' **tt' (i - 1 V' < 2'85> with the same orthonormality conditions (2.84b). F i n a l l y , f o r the e f f e c t i v e operator H A, the eigenvalue equation i s | ( t ) X ^ ) . ( i - 1. .... n A ) , (2.86a) where, 36. and, < X I A ) I ^ A ) > A hy \" L » V - (2'86C) In terms of the eigenvectors of HA, the eigenvectors of the original operator H are ^ [ A > - <1A * D g ^ X i ^ . (2.87) In a l l of these equations, the eigenvalues j» ( i s i» •-•>•§ n A), are exactly the n A eigenvalues of the original operator H corresponding to the eigenvectors f ' j ^ * ( i a 1, n A) • The eigenvalue equations for the effective operators Hg, Gg.,, and Hg, defined in; Sg,, are of the same form as those given above for the corresponding effective operators in S A» Finally, consider the projections P A and Pg, onto the eigenspaces S A, and Sg., respectively. For PA, \" A PA * f l ^ ^ x ^ ^ ^ l \" A » . E ( 1 A • f) Ifi^xfiVl ( 1 A • f f ) (2.88a) Here r E |^ [ ^ x ^ l ^ l * g ( A ) - 1 „ (2.88b) defines an embedding of the inverse of the metric g A i n the n>-dimensional basis space. Similarly, P l . E •|VjB)»«tiB>l i=l 37. - £ ' ( 1 B - f f ) l ^ i f t X ^ i ^ l d B - *>.- (2.89a) where„ X l t i B > < f i f r l * g < B ? M . (2.89b) is an embedding of the inverse of the metric gfi i n the f u l l n-dimensional basis space. 2.2.C Relationships With Other Formulations Many of the quantities defined or derived above have appeared in one form or another in the literature, usually i n connection with the calculation of effective operators i n a perturbation formalism.. The treatment by Friedrichs (1965) of an isolated part of the spectrum of an operator H, is particularly interesting in this regard. Several interrela-tions between the current non-canonical formulation and the more commonly used unitary methods are illustrated by rewriting some of the quantities introduced in that treatment, using the block notation employed here.. Following Friedrichs, the aim; here i s to obtain an expres-sion for a projection operator P^ onto a space spanned by a set of eigenvectors which correspond to an isolated part of the spectrum of some perturbed operator H.. Rather than requiring that the projection P- be orthogonal (that is,, that the operator Fc be hermitian), or ex p l i c i t l y idempotent, i t i s 38:. required only that P€ Po \" V Po P€ * Po' (2.90) where P Q Is a projection onto the corresponding eigenspace of the unperturbed operator tt • These linear conditions, (2.90), imply idempotency, P 2 . p ep 0p e . P , P 0 - p €. and OJ O € O O € O thus verifying that P Q and P^ are projections. However, by themselves, they do not imply that P^ * P €, or that P* = P Q. Equations (2.90) represent the minimal conditions for P £ to be a projection, without prescribing any information about the internal structure of i t s image space. I n a basis adapted to the solution of the zero order problem, that i s , with the matrix representation, Po \" ( 2 . 9 D 0 0_ where the subscript A denotes the space spanned by the zero) order eigenvectors of interest, the form of the matrix repre-sentation of Pg is restricted by (2.90) to 0 0 (2.92) where? f 6 Is a matrix undetermined by (2.90). It is now possible to define mappings, Ug(SQ->Se) and l£(S,->S V. between the spaces S- spanned by the eigenfunctions t c O t 39. <3f the perturbed operator, and the space S Q spanned by the corresponding eigenfunctions of the unperturbed operator. Im terms of the projections P Q and P €, these mappings are U* = H ( P r P 0 ) . (2.93) as given by F r i e d r i c h s . I t then follows that the operator = U* H € U~ r (2.94) i s from S_ to S , but has the same spectrum as Hct, the O) o t F perturbed operator. That i s , H g i s an e f f e c t i v e operator i n the space S . + In the matrix n o t a t i o n introduced above, the mappings U~ are A 0 I, (2.95) i where the subscript B) denotes the space of a l l eigenvectors of H Q except those of i n t e r e s t . Thus, i n the no t a t i o n developed i n the previous sections, H AB D( f e ) ftB?(f€) (2.96) I t i s possible to define a new set of unitary mappings, U, , which map between S_ and S . and vice versa, as € t o U +1 \"A ; f . (2.97) I € \"B|_ Using (2.97) instead of (2.95) i n eq. (2.94),, a new trans-formed perturbed operator i s obtained, 40. (2.98) which is self-adjoint. The operators G A and Gg are given by eqs. (2.67b) and (2.68b). These results are in accord with the fact that the non-selfadjointness of the operators H A and Hfi, introduced im the previous section, i s associated with the fact that the mappings between the two spaces and Sg are, not unitary (that i s , they do not leave the inner product unchanged). We point out that H g is block diagonal when the matrix block f € of U*' satisfies D(f €) = 0 r eq. (2.16). It is inter-+ esting to note that choosing the matrix block f g in U^ of eq. (2.95) to 1 satisfy (2.l6)„ is equivalent to a partial reduction of H>€ toward the upper Hessenberg form, the result of a non-unitary procedure used in numerical matrix diagonali-zation. However, H g is not exactly upper Hessenberg even i f D;(f€) vanishes, because the diagonal blocks of H^ and Hg, are not upper triangular i n general. Finally, note that Friedrichs introduces an operator (P^Pg)\" 1, which is defined only irr the image space S Q of P Q. In the matrix notation used abover P J P € = lk * f j f 6 = ( g i ) A . U.99) Thus* the orthogonal, projection onto S^ ,, given by Friedrichs as G A ( f e ) D ( f € ) T D(f €) G B ( f € ) 41. is written in matrix notation* here as ,-1 P = linr fe a-0 (g c) - l o 0 €'A €'A 0 0 a (2.100) which is identical to the projection P A of eq. (2.10). Finally, we also point out that the operator H, defined by symmetrical orthogonalization i n eqs. (2.74) and (2.76), coincides with operators of Sz.-Nagy (1946/47? see also Riesz and Sz.-Nagy, 1955m §136),, Primas (1961, 1963)r and also? Kato (1966, Remark 4.4 of chapter 2)» 42. 2.3 Generalization to a Nonorthonormal Basis Set The formalism presented i n the f i r s t part of t h i s chapter can e a s i l y he generalized to the s i t u a t i o n i n which the basis functions 0^, ( i * 1. •»•>,, ra)* being used, are not orthonormal, Ira t h i s case, the eigenvalue equation; has the form H X - S X E, (2.101a)) with normalization! X'S X - i n „ (2.101b) where the elements of the matrix S are the inner products of the basis functions, S l j 5 - <0 il0 j>. The p a r t i t i o n i n g of the basis set, and of the eigenvectors of H into two sets of dimensions n A and ng:, respectively, i s car r i e d out exactly as before, leading to eq. (2.2), X = B AA BK = T X , where f and h are again formally given by (2.3). However, as a r e s u l t of the more complicated normalization condition (2.101b), the simple r e l a t i o n (2.4) i s now replaced by |;).. (2.102) h = -*