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Ideas of Quantum Chemistry P27

Ideas of Quantum Chemistry P27 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 226 6. Separation of Electronic and Nuclear Motions these terms11 hh- R kfk - hh- - hh- VR Rfk NR Mfk 2ft 2ft 2ft L J hh r . . . . z. . - 2ft R k Rfk k Rfk R k fk R k R fk h2 2 R k Rfk k Rfk R k fk - 6.12 2ft After inserting the result into l H Mkfkf e and recalling eq. 6.5 we have 2V. . h2 - Ml R k e Rfk Wl I k e h h2 7k- R 2l fk l H k efk -ft e 1 - Skl --- l R k e Rfk - Skl Rfk ft 2ft Hlkfk 6.13 with H lk fi H k e. We obtain the following form of 6.11 E0fl É 1 - Skl - Wl Wk e Rfk - Skl hfk H f Efl. ft 2ft we have profited from the equality fik VRfik e 0 which follows from the differentiation of the normalization condition12 for the function fik Non-adiabatic nuclear motion Grouping all the terms with fl on the left-hand side we obtain a set of N equations 11We use the relation Ar Vr 2. 12We assume that the phase of the wave function fik r R does not depend on R i.e. fik r R j k r R exp i where fik is a real function and R . This immediately gives fiklVR kle ij klVRifik e which is zero from differentiating the normalization condition. Indeed the normalization condition f fik d e 1. Hence Vr f fik d e 0 or 2 f fikVRfik d e 0. Without this approximation we will surely have trouble. 6.3 Adiabatic approximation 227 r h2 i N - Ar E0 R H R - E ft - lkfk 6.14 L E J k l for l 1 2 . N with the non-adiabatic coupling operators h lk - f Vr fk R H k. 6.15 E Note that the operator Hjk depends on the length of the vector R but not on its direction.13 Eq. 6.14 is equivalent to the Schrödinger equation. Eqs. 6.14 and 6.15 have been derived under the assumption that fk of eq. 6.7 satisfy 6.8 . If instead of fk r R we use a generally non-orthogonal complete set if k r R in 6.7 eqs. 6.14 and 6.15 would change to r h2 i N - Ar Ei R H R - E ft - ikfk 6.16 L E J k l for l 1 2 . N with the non-adiabatic coupling operators lk ----- f 11 Rif k e R Hik f l f k e y Ar E 2E 6.17 TV and El R fl Hofl e. 6.3 ADIABATIC APPROXIMATION If the curves E0 R for different l are well separated on the .