Solid State Physics. Philip Hofmann. Читать онлайн. Newlib. NEWLIB.NET

Автор: Philip Hofmann
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Физика
Год издания: 0
isbn: 9783527837267
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the nuclei fixed at these positions and denote the distance in between them by upper R equals StartAbsoluteValue bold upper R Subscript normal upper B Baseline minus bold upper R Subscript normal upper A Baseline EndAbsoluteValue. The Hamiltonian for the hydrogen molecule can then be written as

      where bold r 1 and bold r 2 are the coordinates of the electrons belonging to nucleus A and B, respectively. The first two terms describe the kinetic energies of the two electrons. The operators nabla Subscript 1 Superscript 2 and nabla Subscript 2 Superscript 2 act only on the coordinates bold r 1 and bold r 2, respectively. The electrostatic term contains the repulsion between the two nuclei and the repulsion between the two electrons as well as the attraction between each electron and each nucleus.

      Calculating the energy eigenvalues and wave functions for this Hamiltonian is a formidable problem, mostly because of the interactions between the two electrons (the second term in the curly brackets). We shall return to this problem later. For now, we exploit the fact that the essence of covalent bonding can already be understood by considering just one electron, i.e. by simplifying Eq. (2.3) so that it describes the normal upper H 2 Superscript plusmolecular ion:

      This Hamiltonian is that of an electron moving in the Coulomb potential of two protons separated by a distance upper R. The electrostatic repulsion of the nuclei (the 1 slash upper R term) does not depend on the position of the electron and just leads to an energy offset that could be treated separately. We choose to leave it in the Hamiltonian, though, as we want to inspect the dependence of the resulting energy levels on upper R later.

      We can calculate an approximate ground‐state solution to the Schrödinger equation upper H prime upper Psi left-parenthesis bold r right-parenthesis equals upper E prime upper Psi left-parenthesis bold r right-parenthesis by writing upper Psi left-parenthesis bold r right-parenthesis as a linear combination of the atomic 1s wave functions of the two atoms, phi Subscript normal upper A Baseline left-parenthesis bold r right-parenthesis and phi Subscript normal upper B Baseline left-parenthesis bold r right-parenthesis. This approach is commonly known as linear combination of atomic orbitals (LCAO). Our ansatz is thus

      (2.5)upper H prime upper Psi left-parenthesis bold r right-parenthesis equals upper H prime left-parenthesis c 1 phi Subscript normal upper A Baseline left-parenthesis bold r right-parenthesis plus c 2 phi Subscript normal upper B Baseline left-parenthesis bold r right-parenthesis right-parenthesis equals upper E prime left-parenthesis c 1 phi Subscript normal upper A Baseline left-parenthesis bold r right-parenthesis plus c 2 phi Subscript normal upper B Baseline left-parenthesis bold r right-parenthesis right-parenthesis comma

      where c 1 and c 2 are constants. Multiplying this equation from the left with either phi Subscript normal upper A Superscript asterisk Baseline left-parenthesis bold r right-parenthesis or phi Subscript normal upper B Superscript asterisk Baseline left-parenthesis bold r right-parenthesis and integrating gives two algebraic equations

      where we have introduced the so‐called overlap integral upper S equals integral phi Subscript normal upper A Superscript asterisk Baseline left-parenthesis bold r right-parenthesis phi Subscript normal upper B Baseline left-parenthesis bold r right-parenthesis normal d bold r, as well as the abbreviations upper H prime Subscript AA Baseline equals integral phi Subscript normal upper A Superscript asterisk Baseline left-parenthesis bold r right-parenthesis upper H prime phi Subscript normal upper A Baseline left-parenthesis bold r right-parenthesis normal d bold r and correspondingly for upper H prime Subscript BB Baseline comma upper H prime Subscript BA and upper H prime Subscript AB. As the two nuclei at bold upper R Subscript normal upper A and bold upper R Subscript normal upper B are completely equivalent, we can simplify this by noticing that Скачать книгу