Molecular 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). 

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... 

The statement applies not only to chemical equilibrium but also to phase equilibrium. It is obviously true that it also applies to multiple substitutions. Classically isotopes cannot be separated (enriched or depleted) in one molecular species (or phase) from another species (or phase) by chemical equilibrium processes. Statements of this truth appeared clearly in the early chemical literature. The previously derived Equation 4.80 leads to exactly the same conclusion but that equation is limited to the case of an ideal gas in the rigid rotor harmonic oscillator approximation. The present conclusion about isotope effects in classical mechanics is stronger. It only requires the Born-Oppenheimer approximation. 

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... 

Phenomenological treatments which approximate the molecular potential field (Born-Oppenheimer approximation) by a series of classical energy equations and adjustable parameters. These treatments may be called classical mechanical only in the sense that harmonic force-field expressions stemming from vibrational analysis methods are often introduced, though strictly speaking one is free to select any set of functions that reproduces the experimental data whitin chosen limits of accuracy. 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

There are two reasons why so much is unknown. First, at high densities three (and even four) body forces are important. This is particularly so when chemically reactive atoms are present. Then, even for two-body forces, the strongly repulsive regime is not well understood and, in addition, close in, as one approaches the united atom limit, there is considerable promotion of molecular orbitals. This is a universal mechanism for electronic excitation which means a breakdown of the Born-Oppenheimer approximation for close collisions. 

All aspects of molecular shape and size are fully reflected by the molecular electron density distribution. A molecule is an arrangement of atomic nuclei surrounded by a fuzzy electron density cloud. Within the Born-Oppenheimer approximation, the location of the maxima of the density function, the actual local maximum values, and the shape of the electronic density distribution near these maxima are fully sufficient to deduce the type and relative arrangement of the nuclei within the molecule. Consequently, the electronic density itself contains all information about the molecule. As follows from the fundamental relationships of quantum mechanics, the electronic density and, in a less spectacular way, the nuclear distribution are both subject to the Heisenberg uncertainty relationship. The profound influence of quantum-mechanical uncertainty at the molecular level raises important questions concerning the legitimacy of using macroscopic analogies and concepts for the description of molecular properties. ... 

Some theoretical purists tend to view molecular mechanics calculations as merely a collection of empirical equations or as an interpolative recipe that has very little theoretical Justification. It should be understood, however, that molecular mechanics is not an ad hoc approach. As previously described, the Born-Oppenheimer approximation allows the division of the Schrodinger equation into electronic and nuclear parts, which allows one to study the motions of electrons and nuclei independently. From the molecular mechanics perspective, the positions of the nuclei are solved explicitly via Eq. (2). Whereas in quantum mechanics one solves, which describes the electronic behavior, in molecular mechanics one explicitly focuses on the various atomic interactions. The electronic system is implicitly taken into account through judicious parametrization of the carefully selected potential energy functions. 

The Born-Oppenheimer approximation was developed in 1927 by the physicists Max Born (German) and J. Robert Oppenheimer (American), just one year after Schrodinger presented his quantum treatment of the hydrogen atom. This approximation method is the foundation for all of molecular quantum mechanics, so you should become familiar with it. The basic idea of the Born-Oppenheimer approximation is simple because the nuclei are so much more massive than the electrons, they can be considered fixed for many periods of electronic motion. Let s see if this is a reasonable approximation. Using H2 as a specific example, we estimate the velocity of the electrons to be roughly the same as that of an electron in the ground state (w = 1) of the hydrogen atom. From the Bohr formula, V = we calculate an electron velocity of 2.2 X 10 m sec . We... 

The idea of an effective potential function between the nuclei in a molecule can be formulated empirically (as we have done in Section 3.5 and Fig. 3.9), but it can be defined precisely and related to molecular parameters R, D and Dq only through quantum mechanics. The fundamental significance of the Born-Oppenheimer approximation is its separation of electronic and nuclear motions, which leads directly to the effective potential function. 

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