pchemII.lecture26.Many-Electron_Atoms

pchemII.lecture26.Many-Electron_Atoms - Page 1 of 10 26....

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Page 1 of 10 Scott Kirkby Last revised: 17 March 2008 26. Many-Electron Atoms Suggested Reading: Chapter 13.4 of the text. Introduction: All the systems we have examined up to this point are sufficiently simple that we have been able to obtain rigorous analytical solutions for the eigenfunctions and eigenvalues of the Schrödinger equation describing them This happy state of affairs comes to an abrupt halt when we begin to investigate atoms and molecules containing more than one electron. From this point onward, we shall have to be content with approximate solutions. However, such solutions can often be as accurate as experimental measurement if enough effort is put into the calculation. The fact that things are about to become mush more difficult should not be a cause for concern. Consider an atom that contains a nucleus of mass m n with a positive charge of +Ze and N electrons. For this system, we have a total of N+1 particles, each of which can have kinetic energy. Consequently, we expect the quantum mechanical Hamiltonian to contain N+1 kinetic energy terms of the form (26-1) K.E. term h 2 2 m ------- 2 =
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Physcal Chemistry I Lecture # 26 Page 2 of 10 Scott Kirkby Last revised: 17 March 2008 Each of the electrons will be attracted to the positively charged nucleus. The attractive coulombic interactions will lead to N terms in the potential of the general form (26-2) where ti is the distance of electron i from the nucleus. In writing the coulombic potential in this form, we are using electrostatic units to avoid the factor 4 πε 0 in the denominator. This is the same type of potential term we have for a hydrogen- like atom or ion. Finally, each pair of electrons with the same charge will experience a repulsive interaction that produces a potential term whose form is e 2 /r ij , where r ij is the distance between electrons i and j. The number of such pairwise repulsive interactions is the number of combinations of N objects (electrons) taken two at a time (26-3) Thus, if we are studying the lithium atom, which has three electrons, we find that there are 3(2)/2 = 3 electron-electron repulsion terms in the potential. For sodium, with 11 electrons, the number is 11(10)/2 = 55 terms. Consequently, the Hamiltonian for the sodium atoms will contain 12 kinetic energy terms, 11 nuclear-electron attraction terms, and 55 electron-electron repulsion terms P. E. term Ze 2 r i -------- = N 2 ⎝⎠ ⎛⎞ N ! N 2 () !2! ------------------------ = NN 1 2 --------------------- =
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Physcal Chemistry I Lecture # 26 Page 3 of 10 Scott Kirkby Last revised: 17 March 2008 With the preceding considerations in mind, we see that the Hamiltonian for an N-electron atom is (26-4) Before proceeding, we need to examine the way the electron-electron repulsions are included in the above equation. Let us consider the case of a lithium atom, with three electrons that we denote as 1, 2, and 3. This atoms
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This note was uploaded on 04/19/2008 for the course CHEM PCHEM taught by Professor Kirkby during the Spring '08 term at East Tennessee State University.

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pchemII.lecture26.Many-Electron_Atoms - Page 1 of 10 26....

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