Schrodinger equation molecular spectroscopy

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

Fig. 5.1 Sample IJs) curves for various vibrational states of carbon monosulfide, C = S. These curves were calculated2 in accordance with Eq. (5.2), using i )y(r) functions obtained by solving Schrodinger s equation with an experimental potential energy surface derived from molecular spectroscopy.

A rigged BO approach is developed and used to describe a chemical system calculated with present day advanced electronic methods. Chemical species are determined by electronic wave functions that are independent from the nuclear configuration space. This is the fundamental hypothesis [11]. Boundary conditions in the global electronic wave function are introduced via the solution of electronic Schrodinger equations for systems of external Coulomb sources (Cf. Eq.(8)). The associated stationary arrangement of external Coulomb sources allows for the introduction of molecular frames. This approach naturally leads to a state-to-state description particularly useful in gas phase reactions. A chemical reaction is described as if it were an electronic spectroscopy event or series of events. 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Schrodinger equation is still formidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions form the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

Since its eigenvalues correspond to the allowed energy states of a quantum-mechanical system, the time-independent Schrodinger equation plays an important role in the theoretical foundation of atomic and molecular spectroscopy. For cases of chemical interest, the equation is always easy to write down but impossible to solve exactly. Approximation techniques are needed for the application of quantum mechanics to atoms and molecules. The purpose of this subsection is to outline two distinct procedures—the variational principle and perturbation theory— that form the theoretical basis for most methods used to approximate solutions to the Schrodinger equation. Although some tangible connections are made with ideas of quantum chemistry and the independent-particle approximation, the presentation in the next two sections (and example problem) is intended to be entirely general so that the scope of applicability of these approaches is not underestimated by the reader. 

Equation (13.25) is the same as the Schrodinger equation for a one-dimensional harmonic oscillator with coordinate x, mass (i, potential energy kgX, and energy eigenvalues Ejm — U(Rg) — J J + Vfi /2tt.R. [The boundary conditions for (13.25) and (4.34) are not the same, but this difference is unimportant and can be ignored Levine, Molecular Spectroscopy, p. 147).] We can therefore set the terms in brackets in (13.25) equal to the harmonic-oscillator eigenvalues, and we have... 

Equation 8.25 gives the solution to the energy levels of the simple harmonic oscillator, a model for the vibrational mechanics of a chemical bond. This is only a model, however, and we know that it doesn t succeed under all conditions. Often in vibrational spectroscopy we look no further than the lowest excited state, V = 1, and in that case Eq. 8.25 is usually adequate. It predicts rather well, for example, how the transition energy depends on the atomic masses. However, detailed studies of molecular dynamics and interactions demand a more general approach to the vibrational Schrodinger equation. In this section, we look at how the harmonic oscillator model fails and what we can do about it. 

Spatial and spin symmetry are properties of the solution of the (non-relativistic) Schrodinger equation and therefore of molecular states. Besides providing convenient labels, a state s symmetry properties provide basic information about its spectroscopy and bonding properties. State symmetry is a many-electron property, but the symmetry of one-electron wavefunctions (molecular orbitals, MOs) is also defined within most contexts of approximations to many-electron wavefunctions. MO symmetries are additionally useful in understanding spectroscopic and bonding properties, as well as molecular rearrangements through the Woodward-Hojfmann rules. 

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