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Part 2 Atomic Orbital Based Response Theory The first term is simply the geometrical gradient of the ground state. In ref. 66 this was shown to be E 0 x = 2Tr x + Tr x ( ) + Tr x + hnuc x Dh DG D D F . (2.103) The other terms are found as the derivative of the matrix expressions in Eq. (2.91) and (2.92) f † [2] ∂b E b ∂x f † [2] f f ( ( )) =− Tr F + G D ⎡ , , ⎤ −Tr ⎡ , , ⎤ ⎣ b D b ⎦ F ⎣ b D b ⎦ x x f f † f x f † ⎡⎣ ⎤⎦ ⎡ ⎤ S S ⎣ ⎦S −Tr F⎡⎡⎣ , ⎤⎦ x , ⎤ Tr ⎡⎡ ⎣ , ⎤⎦ , ⎤ ⎣ b D b ⎦ F ⎣ b D b ⎦ f f † f f † − S S S † ( ⎡ ⎤ ) ⎡ ⎤ S S f x f † f † ( ⎡ ⎤ )( ⎡ ⎤ ⎡ ⎤ x ) x f f − Tr G ⎣b , D⎦ ⎣D, b ⎦ − 2Tr G ⎣b , D⎦ ⎣D , b ⎦ + ⎣D, b ⎦ ∂b S b f † x f − ω = −ωTr b S ⎡ , ⎤ ∂x ⎣D b ⎦ S S S S S ( ⎡ ⎤ ⎡ ⎤ x ⎡ ⎤ ) f † x f f f x ⎣ ⎦S ⎣ ⎦S ⎣ ⎦S − ω Tr b S D , b S+ D, b S+ D, b S S x S (2.104) (2.105) ∂( FDS −SDF ) x x x x A − X = − 2X⎡ + ( ) + + ⎤ ∂x ⎣F DS G D DS FD S FDS ⎦ , (2.106) where F x = h x + G x (D). Collecting the various terms we obtain f ∂E ∂x f f † x f † x f ( D ⎡ ⎤ ⎣ ⎡⎣b D⎤⎦ b [ ] S ⎦ D X S ) h ⎡ ⎤ S ( ⎡ ⎤ S ) S ⎣D b ⎦ G ⎣b D⎦ f f † x x ( D ⎡⎡ ⎣b D⎤⎦ b ⎤ [ D X] ) G D hnuc = Tr 2 − , , − , −Tr , , + Tr − ⎣ , , ⎦ − , ( ) + S S S x f f † x f † f † f DG( ⎡ ⎤ ⎣ ⎡⎣b D⎤⎦ b ) ( x S ⎦ ⎡ S S ) ( S ) S ⎣D b ⎤⎦ ⎡⎣Db ⎤⎦ G ⎡⎣b D⎤⎦ x x DG( [ DX] ) ( ⎡ ⎤ [ ] x S ⎣D X⎦ DX S S ) F f x ( ⎡ f † † † ⎡⎣ b D ⎤ , ⎤ ⎡ f f , x , ⎤ ⎡ f f , , ⎤ ⎣ ⎦ b S ⎦ + S S x ) S ⎣⎣ ⎡b D⎦⎤ b ⎦ + S ⎣⎣ ⎡b D⎦⎤ b ⎦ F S f † f x f f x Tr b S( ⎡b , D ⎤ S ⎡b , D⎤ x S ⎡b , D⎤ S ) −Tr , , − 2Tr , + , , −Tr , − Tr , + , − Tr , + ω f ⎣ ⎦ + ⎣ ⎦ + ⎣ ⎦ f † x f + ω f Tr b S ⎡⎣b , D⎤⎦ S, G b D b , ( [ , ] ) where ( ⎡ f f † , , ⎤ ⎣ ⎡⎣ ⎤⎦S ⎦ ) S G x x f (D), ( ⎡ , ⎤ ) S S S S f G D X , ( ⎡ , ⎤ ) (2.107) G S ⎣ b D ⎦ and F can be evaluated, whereas S G ⎣ b D ⎦ , h x and nuc x h have to be evaluated for each geometrical perturbation. S Note that no two-electron integrals are represented explicitly, in order to obtain the best performance – e.g. for linear scaling codes - no reference should be made to four-index integrals. 2.4.4 The First-order Excited State Properties The expression for the first-order one-electron excited state properties for perturbation independent basis sets is obtained from the expression for the excited state gradient by omitting all two-electron derivative terms, as well as all terms involving the derivative of the overlap matrix 74
Test Calculations ( ⎡ † ⎡⎣ ⎤⎦ ⎤ [ ] ) x 2Tr x Tr f , , f , x x = − ⎣ S ⎦ − + S S nuc f h f Dh b D b D X h h . (2.108) The first and last terms in Eq. (2.108) correspond to the ground state first order property as seen from Eq. (2.103). 2.5 Test Calculations To illustrate the possibilities of an AO response solver in connection with our SCF optimization program, test calculations have been carried out on problematic cases from the first part of the thesis. The lowest excitation energy and the average polarizability, both static and in a field with ω = 0.03a.u., have been found for the zinc complex in Fig. 1.3 and the rhodium complex in Fig. 1.33. The levels of theory chosen are those where DIIS could not optimize the reference state, namely LDA/6-31G for the zinc complex and HF/AhlrichsVDZ with STO-3G on the rhodium for the rhodium complex. Table 2-1 Ground state properties obtained with our AO response solver. All numbers are in a.u. The average polarizability Excitation static ω = 0.03 energy Rhodium complex HF/AhrichsVDZ 170.598 173.349 0.0938 Zinc complex LDA/6-31G 161.406 162.517 0.0713 The basis sets applied in the test calculations are not satisfactory for serious polarizability calculations, and the numbers only demonstrate the perspectives of the AO response solver in combination with the SCF optimization algorithms described in Part 1. When the solver is fully implemented in the AO basis, we will be able to obtain molecular properties for large complex molecules in a routine manner. The implementation of the excited state gradient is a work in progress. So far we have implemented calculation of first-order one-electron properties of the excited state for perturbation independent basis sets as described in Section 2.4.4. The excited state dipole moment of the Rhodium complex from above has been found as Rh Cl µ = 5.960a.u. Again it should be noted that the basis set is insufficient for this type of calculation. This is only to demonstrate that it can be done. 75
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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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