The Molecular Schrodinger Equation

In this equation, atomic units have been used in which H = 4n8o = 1 and consequently energy is measured in Hartree (1 Eh = 27.2107 eV) and length in Bohrs (1 Bohr = 0.529... 10 ° m). The solutions to the BO Schrodinger equation are the many-electron wavefunctions /(xi,. ..,Xw) (1 = 0, 1,. .., oo) and their associated energies Ej. The square of the many-electron wavefunction provides the probability density for finding the electrons at positions ri.rA with spins (Ti,. ..,(7Af(x,- = (ri,(Ti)). 

The individual terms in (5.2) and (5.3) represent the nuclear-nuclear repulsion, the electronic kinetic energy, the electron-nuclear attraction, and the electron-electron repulsion, respectively. Thus, the BO Hamiltonian is of treacherous simplicity it merely contains the pairwise electrostatic interactions between the charged particles together with the kinetic energy of the electrons. Yet, the BO Hamiltonian provides a highly accurate description of molecules. Unless very heavy elements are involved, the exact solutions of the BO Hamiltonian allows for the prediction of molecular phenomena with spectroscopic accuracy that is 

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Substitution of Eq. (3) into the molecular Schrodinger equation leads to a system of coupled equations in a coupled multistate electronic manifold... 

The molecular Schrodinger equation can be solved exacdy for the case of Hj when VAB is simply the sum of two hydrogen ion potentials. In general, however, an exact solution is not possible. Following the well-worn tracks of MO theory we look instead for an approximate solution that is given by some linear combination of atomic orbitals (LCAO). Considering the AB dimer illustrated in Fig. 3.1 we write... 

The cylindrical symmetry about the inter-nuclear axis leads to the solutions of the molecular Schrodinger equation, eqn (3.3), having either a or character. Taking the z axis along the axis of the molecule, the a eigenfunctions will comprise linear combinations of the , , and atomic orbitals so that we can write the molecular orbital as... 

The development of approximate methods of applying quantum mechanics to molecules and molecular systems leads immediately to the introduction of models. Seldom is discussion of the properties of a given molecule or molecular complex based on the direct solution of the molecular Schrodinger equation (or its relativistic generalization). Instead, approximations are invoked which not only make the computation tractable but also introduce concepts and models upon which our interpretation of the properties of the molecule are based. 

The first two terms are the molecular Hamiltonian and the radiation field Hamiltonian. The molecular Schrodinger equation for the first term in (5.2) is assumed solved, with known eigenvalues and eigenfunctions. Solutions for the second term in (3.4) in vacuo are taken in second-quantized form. Hint can be taken in minimal-coupling form (5.3) allowing for the variation of the radiation field over the extent of the molecule,... 

V.I. Pupyshev, A.V. Scherbinin, N.F. Stepanov, The Kirkwood-Buckingham variational method and the boundary value problems for the molecular Schrodinger equation, J. Math. Phys. 38 (11) (1997) 5626-5633. 

In principle, most physical and all chemical process will be represented as a change among the exact solutions of the molecular Schrodinger equation ... 

For these studies we will use the molecular Schrodinger equation. 

The irreducible representations of a symmetry group of a molecule are used to label its energy levels. The way we label the energy levels follows from an examination of the effect of a symmetry operation on the molecular Schrodinger equation. 

The molecular Schrodinger equation can be solved exactly for the case of H, for the hydrogen ion. It is a case of the true covalent bond. Here we have two nuclei repulsing each other, and both exerting a Coulomb attraction on the single electron. 

Ab initio methods solve the molecular Schrodinger equation associated with the molecular Hamiltonian based on different quantum-chemical methodologies that are derived directly from theoretical principles without inclusion of any empirical or semiempirical parameters in the equations. Though rigorously defined on first principles (quantum theory), the solutions from ab initio methods are obtained within an error margin that is qualitatively known beforehand thus all the solutions are approximate to some extent. Due to the expensive computational cost, ab initio methods are rarely used directly to study the physicochemical properties of flotation systems in mineral processing, but their application in developing force fields for molecular mechanics (MM) and MD simulation has been extensively documented. (Cacelli et al. 2004 Cho et al. 2002 Kamiya et al. 

Because the motions of the various particles are correlated through the potential terms of Eq. (2.3), the direct integration of the molecular Schrodinger equation is an extremely difficult task that is only possible, in practice, for the simplest atomic and molecular systems. 

The Molecular Schrodinger Equation The exact molecular TDSE then reads... 

We want to solve the molecular Schrodinger equation. No, really, we do, because the resulting energies and wavefunctions will predict the behavior of molecules in situations that we care about in all branches of chemistry why the molecule has the spectrum it has, whether it forms a liquid or solid or gas, why it reacts the way it does. The molecular Hamiltonian unlocks the whole world of chemistry. But it s a big world, and we re going to travel it in steps, beginning with how the Coulomb force dictates the shape of the potential energy term. 

The goal of this chapter is to describe the quantum states and energies that result from solving the electronic part of the molecular Schrodinger equation. 

Now our job is to find the energies and wavefunctions for that third part of the molecular Schrodinger equation, the rotational term. As usual, we start with the simplest case—which for molecules means diatomics. 

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