Nuclear motion Schrodinger equation principles

The first problem is the subject of modern quantum chemistry. Many efficient methods and computer programs are nowadays available to solve the electronic Schrodinger equation from first principles without using phenomenological or experimental input data (so-called ab initio methods Lowe 1978 Carsky and Urban 1980 Szabo and Ostlund 1982 Schmidtke 1987 Lawley 1987 Bruna and Peyerimhoff 1987 Werner 1987 Shepard 1987 see also Section 1.5). The potentials Vfe(Q) obtained in this way are the input for the solution of the nuclear Schrodinger equation. Solving (2.32) is the subject of spectroscopy (if the motion is bound) and... 

The interaction potential between solvent molecules and between solvent and solute molecules may, in principle, be found by solving the electronic Schrodinger equation, Eq. (3.1), for the solute/solvent system. Thus, the electronic energy (the potential for the nuclear motion) is, for a system with N nuclei, a function of the position vectors of each nucleus, i.e., Un t, r/v). Note that if the whole system is invariant to... 

The adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

Apart from use of experimental values of atomic - rather than nuclear - and electronic masses and of electric charges, the basis of this calculation has an empirical component. The calculation is certainly not made genuinely from first principles or ab initio, firstly because the composition of the basis set is predetermined, by those who have published this basis set [12] and by the authors of Dalton software [11] who have incorporated it, according to its success in reproducing experimentally observable quantities and other calculated properties. Secondly, the solution of Schrodinger s equation is based on a separation of electronic and nuclear motions, essentially with atomic nuclei fixed at relative positions, which is a further empirical imposition on the calculation efforts elsewhere to avoid such an arbitrarily distinct treatment of subatomic particles, even on much simpler molecular systems, have... 

The Born-Oppenheimer principle assumes separation of nuclear and electronic motions in a molecule. The justification in this approximation is that motion of the light electrons is much faster than that of the heavier nuclei, so that electronic and nuclear motions are separable. A formal definition of the Born-Oppenheimer principle can be made by considering the time-independent Schrodinger equation of a molecule, which is of the form... 

In Section 2.1, the electronic problem is formulated, i.e., the problem of describing the motion of electrons in the field of fixed nuclear point charges. This is one of the central problems of quantum chemistry and our sole concern in this book. We begin with the full nonrelativistic time-independent Schrodinger equation and introduce the Born-Oppenheimer approximation. We then discuss a general statement of the Pauli exclusion principle called the antisymmetry principle, which requires that many-electron wave functions must be antisymmetric with respect to the interchange of any two electrons. 

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