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Part 2 Atomic Orbital Based Response Theory AY + FU + BZ = − ω ( ΣY + ΩU + ∆Z ) ( ) T T T T F Y+ GU+ F Z = −ω Ω Y −Ω Z BY + FU + AZ = −ω ( −∆Y −ΩU −ΣZ ), (2.58) which are identical to Eqs. (2.55). It is thus concluded that if X is an eigenvector of Eq. (2.45) with eigenvalue ω, then X P is also an eigenvector with eigenvalue –ω. 2.3 Solving the Response Equations For large systems, the response equations ( ω ) [2] [2] [1] E − S N B ( ω ) = B (2.59) are best solved using iterative algorithms. These algorithms rely on the ability to set up linear transformations. Expressions for E [2] b and S [2] b, where b is a trial vector, have previously been derived. 61 [2] σ = E b (2.60) [2] ρ = S b. (2.61) In each iteration, the response equations are set up and solved in a reduced space. For a reduced space consisting of k trial vectors, the equations can be written as where the reduced matrices are found as ( ω ) [2] [2] RED [1] RED − RED = RED E S X B , (2.62) [2] T [2] T RED ⎦ i j i j ij ⎡ ⎣ E ⎤ = b E b = b σ [2] T [2] T RED ⎦ i j i j ij ⎡ ⎣ S ⎤ = b S b = b ρ [1] T [1] RED ⎦ bi B . i ⎡ ⎣ B ⎤ = (2.63) Normally when this type of iterative procedure is used, the reduced space is extended with one new trial vector in each iteration. However, due to the pairing described in the previous section, the linear transformations of E [2] and S [2] on a trial vector, here exemplified by E [2] b, ⎛ A F B ⎞⎛ Z⎞ ⎛ AZ+ FU+ BY ⎞ [2] ⎜ T T ⎟⎜ ⎟ ⎜ T T E b = F G F U = F Z+ GU+ F Y ⎟ = σ , (2.64) ⎟⎜ ⎜ ⎟⎜ ⎟ ⎜ + + ⎟ ⎝ B F A ⎠⎝Y⎠ ⎝ BZ FU AY ⎠ may be obtained directly for the paired trial vector as well ⎛ A F B ⎞⎛Y⎞ ⎛ AY+ FU+ BZ ⎞ [2] P ⎜ T T ⎟⎜ ⎟ ⎜ T T ⎟ P E b = F G F U = F Y+ GU+ F Z = σ . (2.65) ⎟⎜ ⎜ ⎟⎜ ⎟ ⎜ + + ⎟ ⎝ B F A ⎠⎝ Z⎠ ⎝ BY FU AZ ⎠ 68
Solving the Response Equations The reduced space is therefore extended with both vectors without additional cost. Furthermore, when a trial vector and its paired counterpart are simultaneously added to the reduced space, the paired structure of the response equations is preserved. With this structure preserved, the eigenvalues in the reduced space will also be real and paired, and the lowest eigenvalue will monotonically decrease towards the converged value as the reduced space is increased. 64 The solution vector in the reduced space X RED , can be expanded in the basis of trial vectors to express the solution vector in the full space k B . (2.66) N = ∑ i= 1 RED ( X i bi ) The residual can then be found as k ( ω ) R = E − S N −B k ∑ [2] [2] B [1] = X ( σ −ωρ ) −B i= 1 RED [1] i i i . (2.67) If the norm of the residual is smaller than some specified tolerance, the iterative procedure is ended and the converged solution vector has been found B B N ( ω ) = N . (2.68) If the residual is too large, a new trial vector may be generated from the residual, preferably with a preconditioner A to speed up the convergence k+ 1 = −1 b A R . (2.69) The reduced space is then extended with b k+1 and bk + 2 = b k + 1 and Eq. (2.62) is set up and solved again, establishing the iterative procedure. 2.3.1 Preconditioning As mentioned above, the residual found in each iteration should be preconditioned to obtain an effective solver. As a consequence of the strict AO formulation, the electronic Hessian has no diagonal dominance as was the case in the MO basis. This makes preconditioning a challenge. So far, this problem has not been solved in our SCF response solver. Instead, a transformation is made to the MO basis, where the preconditioning is carried out in the usual way using the orbital eigenvalue differences, k P MO T ⎣⎡b + 1 ⎦⎤ = ⎣⎡C RkC ⎦⎤ ( εa −εi ), (2.70) k ai ai 69
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PhD Thesis Optimization of Densitie
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Contents Preface ..................
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Summary............................
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List of Publications This thesis in
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Part 1 Improving Self-consistent Fi
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