Quantum mechanics Born-Oppenheimer approximation

Both molecular and quantum mechanics methods rely on the Born-Oppenheimer approximation. In quantum mechanics, the Schrodinger equation (1) gives the wave functions and energies of a molecule. 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

In this section the Born-Oppenheimer approximation will be presented in what is necessarily a very simplified form. It has already been introduced without justification in Section 6.5. It is certainly the most important - and most satisfactory - approximation in quantum mechanics, although its rigorous derivation is far beyond the level of this book. Consider, therefore, the Mowing argument... 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

One branch of chemistry where the use of quantum mechanics is an absolute necessity is molecular spectroscopy. The topic is interaction between electromagnetic waves and molecular matter. The major assumption is that nuclear and electronic motion can effectively be separated according to the Born-Oppenheimer approximation, to be studied in more detail later on. The type of interaction depends on the wavelength, or frequency of the radiation which is commonly used to identify characteristic regions in the total spectrum, ranging from radio waves to 7-rays. 

The problem, as Woolley addressed it, is that quantum mechanical calculations employ the fixed, or "clamped," nucleus approximation (the Born-Oppenheimer approximation) in which nuclei are treated as classical particles confined to "equilibrium" positions. Woolley claims that a quantum mechanical calculation carried out completely from first principles, without such an approximation, yields no recognizable molecular structure and that the maintenance of "molecular structure" must therefore be a product not of an isolated molecule but of the action of the molecule functioning over time in its environment.47... 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

In principal one can calculate the electronic energy as a function of the Cartesian coordinates of the three atomic nuclei of the ground state of this system using the methods of quantum mechanics (see Chapter 2). (In subsequent discussion, the terms coordinates of nuclei and coordinates of atoms will be used interchangeably.) By analogy with the discussion in Chapter 2, this function, within the Born-Oppenheimer approximation, is not only the potential energy surface on which the reactant and product molecules rotate and vibrate, but is also the potential... 

At this stage we are at the very beginning of development, implementation, and application of methods for quantum-mechanical calculations of molecular systems without assuming the Born-Oppenheimer approximation. So far we have done several calculations of ground and excited states of small diatomic molecules, extending them beyond two-electron systems and some preliminary calculations on triatomic systems. In the non-BO works, we have used three different correlated Gaussian basis sets. The simplest one without r,y premultipliers (4)j = exp[—r (A t (8> Is) "]) was used in atomic calculations the basis with premultipliers in the form of powers of rj exp[—r (Aj (8> /sjr])... 

Almost all studies of quantum mechanical problems involve some attention to many-body effects. The simplest such cases are solving the Schrodinger equation for helium or hydrogen molecular ions, or the Born— Oppenheimer approximation. There is a wealth of experience tackling such problems and experimental observations of the relevant energy levels provides a convenient and accurate method of checking the correctness of these many-body calculations. 

Having developed the mathematics of group theory, we now apply it to molecular quantum mechanics. As usual, we use the Born-Oppenheimer approximation. 

The material model consists of a large assembly of molecules, each well characterized and interacting according to the theory of noncovalent molecular interactions. Within this framework, no dissociation processes, such as those inherently present in water, nor other covalent processes are considered. This material model may be described at different mathematical levels. We start by considering a full quantum mechanical (QM) description in the Born-Oppenheimer approximation and limited to the electronic ground state. The Hamiltonian in the interaction form may be written as ... 

Whereas the quantum-mechanical molecular Hamiltonian is indeed spherically symmetrical, a simplified virial theorem should apply at the molecular level. However, when applied under the Born-Oppenheimer approximation, which assumes a rigid non-spherical nuclear framework, the virial theorem has no validity at all. No amount of correction factors can overcome this problem. All efforts to analyze the stability of classically structured molecules in terms of cleverly modified virial schemes are a waste of time. This stipulation embraces the bulk of modern bonding theories. 

The classical idea of molecular structure gained its entry into quantum theory on the basis of the Born Oppenheimer approximation, albeit not as a non-classical concept. The B-0 assumption makes a clear distinction between the mechanical behaviour of atomic nuclei and electrons, which obeys quantum laws only for the latter. Any attempt to retrieve chemical structure quantum-mechanically must therefore be based on the analysis of electron charge density. This procedure is supported by crystallographic theory and the assumption that X-rays are scattered on electrons. Extended to the scattering of neutrons it can finally be shown that the atomic distribution in crystalline solids is identical with molecular structures defined by X-ray diffraction. 

The Born-Oppenheimer separation19-22 of the electronic and nuclear motions in molecules is probably the most important approximation ever introduced in molecular quantum mechanics, and will implicitly or explicitly be used in all subsequent sections of this chapter. The Born-Oppenheimer approximation is crucial for modern chemistry. It allows to define in a rigorous way, within the quantum mechanics, such useful chemical concepts like the structure and geometry of molecules, the molecular dipole moment, or the interaction potential. In this approximation one assumes that the electronic motions are much faster than the nuclear... 

Atomic units are used in all equations and all considerations concern non-relativistic quantum mechanics in Born-Oppenheimer approximation. Square brackets, as in E[p] for instance, are used to indicate that the relevant quantity is a functional i.e. the correspondence between a function in real space p = p(r) and a real number (energy in this example). Abbreviations or acronyms denoting different approximate exchange-correlation functionals reflect their common usage in the literature. They are collected in Appendix. Unless specified, the equations are given for the spin-compensated case. 

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