# The Solid Solution Theory of Dyeing by Bancroft W. D.

By Bancroft W. D.

By Bancroft W. D.

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15] (Ec); b) long dashed line: calculated up to second order in full space (E(2)); c) short dashed line: calculated up to second order in the space of orbitals with negative energies (E (2) (ε < 0)); d) dotted line: calculated with a configuration interaction in 1s,2s,2p subspace (Ec, q ) ; e) dashed dotted line: calculated up to second order in 1s,2s,2p subspace (Eq(2)) . CORRELATION ENERGY CONTRIBUTIONS 37 lations assuming that one can define a local gap (|∇ n|/n)2 (where n is the electron density of the system considered) which asymptotically is equal to the ionization potential [28].

DELCHEV ET AL. are not linearly independent. fR ) that produce linearly dependent sets, for which the Grammian is → → → equal to zero: (26) and The groups reduced in this manner are mutual isomor R → R Let us fix a normalized set of R linearly independent orbitals {ϕi (x)} i=1 ∈L1 (we restore the spin variables in the notation). The manifold of all orbital sets - Eqn (25) - forms an orbit: {ψi }i=R 1 induced by the operators (27) R where {ϕi(x)} i= 1 is the orbit-generating set. For this orbit ∂[{ϕi}Ri=1 ] ≡ ∂ϕR , holds the inclusion relation Hence ∂Rϕ is that manifold in L 1 which consists R induced by the eleof all sets of linearly independent orbitals {Ψi([fi (r)];x)}i=1 ments of the group Let R = N and (28) → N N be the N-particle product for a chosen spin-orbital set {ϕi(x)} i=1 ∈ L 1 Antisymmetrizing the expression (28) yields the corresponding Slater determinant: (29) where AN is the antisymmetrizer, LN the antisymmetric N-particle Hilbert space, and SN the subclass of single Slater determinants in LN .

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