By Marco A. Pérez B.

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In terms o f these Η — I vectors, in contradiction t o their linear independence. #1 n 2 v w el 2 N Finally, we normalize the U . , U . , · · * , U. , thereby constructing a unitary matrix whose first column is U . This "SCHMIDT ORTHOGONALIZATION PROCEDURE" shows how t o construct from any set o f linearly independent vectors an orthogonal normalized set in which the KTH unit vector is a linear combination o f just the first K o f the original vectors. I f one starts with Η η-dimensional vectors which form a complete set o f vectors, one obtains a complete ORTHOGONAL system.

For t>. ; this leaves the determinant unaltered. Then set 2 #1 2 3 2 (U. = 0 = a {U. tt. ) v 21 2 and determine a 21 from this. i) + (U. v 2 21 + (u. ) v 2 Next write U. in place of t>. with tt. 3 d determine a and a 31 « . , ) = a (U. ) = a 1? ) v 3 0 = (U. 3) = a (M. ) + (U. ,W. ). 2 32 2 2 2 Proceeding in this way, we finally write U. n U. _ w a n,n-V s o 3 in place o f t). , with n. w w x n nl n2 ^at 0 = (n. J = a (U. „), 0 = (U. „) = a (U. ,1l. ) + (U. , V. ), 2> 0 = (u. _ n v n2 = + V. ^, u.

W e have already assumed that these are mutually orthogonal. 15) *ssWks\ Consequently, there is no longer any reason to assume that the first approxi matrix, the y) are also mutually mation t o 9? is simply ip . I f ( & , ) Ϊ 8 a unitary fc k μ μ kv GROUP T H E O R Y A N D ATOMIC SPECTRA 44 (and, of course, orthogonal to other eigenfunctions, with eigen orthogonal values different from E ) . k (ψίν> = Thus, the the \p' Σ ν ν ' ν'μ' <*μμ' α are kv original ψ*μ) as = < V (Σ \Wkv'> a ° Σ W λ ^νν'^μμ' ν'μ' = suitable Vkv>> basis for Wk^ ν'μ' 1 = Ο °νμ· J the approximation procedure as y) .