Born-Oppenheimer approximation, electronic structure calculations

Using the Born-Oppenheimer approximation, electronic structure calculations are performed at a fixed set of nuclear coordinates, from which the electronic wave functions and energies at that geometry can be obtained. The first and second derivatives of the electronic energies at a series of molecular geometries can be computed and used to find energy minima and to locate TSs on a PES. 

Background Philosophy. Within the framework of the Born-Oppenheimer approximation (JJ ), the solutions of the Schroedin-ger equation, Hf = Ef, introduce the concept of molecular structure and, thereby, the total energy hyperspace provided that the electronic wave function varies only slowly with the nuclear coordinates, electronic energies can be calculated for sets of fixed nuclear positions. The total energies i.e. the sums of electronic energy and the energy due to the electrostatic re-... 

A full theoretical treatment of Pgl within the Born-Oppenheimer approximation involves two stages. First, the potentials V (R), V+(R), and the width T(f ) of the initial state must be calculated in electronic-structure calculations at different values of the internuclear distance R—within the range relevant for an actual collision at a certain collision energy—and then, in the second stage the quantities observable in an experiment (e.g., cross sections, energy and angular distributions of electrons) must be calculated from the functions V+(R), V+(R), and T(f ), taking into account the dynamics of the heavy-particle collisions. 

Schrodinger equations for atoms and molecules use the the sum of the potential and kinetic energies of the electrons and nuclei in a structure as the basis of a description of the three dimensional arangements of electrons about the nucleus. Equations are normally obtained using the Born-Oppenheimer approximation, which considers the nucleus to be stationary with respect to the electrons. This approximation means that one need not consider the kinetic energy of the nuclei in a molecule, which considerably simplifies the calculations. Furthermore, the... 

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.)... 

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. 

Hartree-Fock (HF) theory " is the wavefunction model most often used to describe the electronic structure of atoms and molecules. When the Born-Oppenheimer approximation " can be made, one can find an approximation of the many-electron wavefunction T of a system by a variety of quantum chemical methods. When F is known, one calculates the expectation value A xp of a quantity A from... 

In most electronic-structure calculations, the Born-Oppenheimer approximation is invoked. This leads to flow charts like that of Figure 1, where the electronic properties are determined for a given set of nuclear positions. This is done by solving some kind of time-independent electronic Schrodinger equation,... 

In the approaches we have discussed so far, the Born-Oppenheimer approximation is employed, i.e., the nuclei are kept at some positions and the electronic structure for this structure is calculated self-consistently, after which the positions of the nuclei may be changed., e.g., by using the forces acting on the nuclei. This corresponds to the representation of Figure 1. As an alternative, Car and Parrinello14 suggested to determine the electronic distribution and the nuclear coordinates simultaneously. To this end they constructed an artificial Lagrangian,... 

Section 2 of this chapter notes will be devoted to the framework for separation of the ionic and electronic dynamics through the Born-Oppenheimer approximation. Atomic motion, with forces on the ions at each timestep evaluated through an electronic structure calculation, can then be propagated by Molecular Dynamics simulations, as proposed by first-principle Molecular Dynamics. This allows for a description of the electronic reorganisation following the atomic motion, e.g. bond rearrangements in chemical reactions. 

The structure of approximate reasoning is not simple. Consider the Born-Oppenheim approximation (separability of electronic and nuclear motions due to extreme mass difference), which in application produces "fixed nuclei" Hamiltonians for individual molecules. In assuming a nuclear skeleton, the idealization neatly corresponds to classical conceptions of a molecule containing localized bonds and definite structure. All early quantum calculations, and the vast majority to date, invoke the approximation. In 1978, following decades of quiet assumption, Cambridge chemist R. G. Woolley asserted ... 

This method differs fundamentally from the other calculational methods previously mentioned. The Born-Oppenheimer approximation says that one can separate the nuclear from the electronic motions in a molecule, and the previously discussed quantum mechanical methods have to do with the electronic system, after the nuclear positions have been established (or assumed). To determine structures by such methods, one must repeat the calculation for a number of different nuclear positions, and locate the energy minimum in some way. Unless the structure is known at the outset, one therefore requires not just a single calculation, but many calculations, in order to determine the actual structure. 

Our problem now is to see how this shape arises from the electronic structure of the atoms involved and to find out what we can about wave functions, energies, charge distributions and so on, all of which will be needed for an understanding of the interactions of these molecules in the liquid and solid states. Even using the Born-Oppenheimer approximation, which allows us first to solve the electronic problem with the nuclei fixed and then to use this result to determine the effective potential in which the nuclei move, exact solution of the Schrodinger equation is out of the question. It is possible, however, with relatively little labour, to see how the particular structure of the water molecule comes about and then, by refining this crude model, to calculate relevant quantities quite accurately. 

The molecular mechanics method is used to calculate molecular structures, conformational energies, and other molecular properties using concepts from classical mechanics. Electrons are not explicitly included in the molecular mechanics method, which is justified on the basis of the Born-Oppenheimer approximation stating that the movements of electrons and the nuclei can be separated. Thus, the nuclei may be viewed as moving in an average electronic potential field, and the molecular mechanics method attempts to describe this field by its force field. ... 

In the first part of this chapter, we have reported a brief survey of the computational tools available to evaluate the geometries and relative energies of the stationary points (reactants, transition structures, and reaction intermediates) associated with the elementary steps of any transformation under the Born-Oppenheimer approximation. These computational data permit the calculation of individual kinetic constants associated with each step. In the second part, we have commented on several results obtained in our laboratory. These kinetic calculations involved up to 40 atoms and 160 electrons and in many cases included solvent effects. Under these conditions, the individual rate constants associated with each elementary step were expected not to be very accurate. However, since the error magnitudes along the different reaction coordinates were similar enough, the final relative rate constants reproduced surprisingly well the experimental observations, even when very subtle effects were studied. In addition, the possibility of evaluating each step of the different reactions permitted... 

The Born-Oppenheimer approximation simplifies quantum mechanical calculations by considering the electronic wave function for a structure separately from that of the nuclei Bom, M. Oppenheimer, J. R. Ann. Phys. 1927, 84, 457. 

For you to solve the electronic structure and energy of a molecule, quantum mechanics requires that you formulate a wavefunction P (psi) that describes the distribution of all the electrons within the system. The nuclei are assumed to have relatively small motions and to be essentially fixed in their equilibrium positions (Born-Oppenheimer approximation). The average energy of the system is calculated by using the Schrodinger equations as... 

Considerable advances in the formal development of DFT as well as in its practical application have been made these last few years. In this chapter, we will not attempt to provide a comprehensive review of the subject, but rather provide an introduction to DFT, covering many of the important attributes that pertain to chemical problems. In the next section, we provide some of the basic calculus associated with functionals. Then some early models used in DFT are presented, followed by a formal presentation of DFT and the formalism used in practical applications. Finally, in the last section, we discuss the ability of DFT to compete with WFT in electronic structure calculations. A more detailed and comprehensive discussion of DFT can be found in the 1989 book by Parr and Yang. In what follows, we assume the Born-Oppenheimer nonrelativistic approximation and use atomic units throughout. Since the nuclei are assumed to be fixed in space, we will not explicitly show this dependence in equations and variables. 

Within the Born-Oppenheimer approximation, we still need to know that the nuclear position parameters really correspond to the distances and angles of a classical molecular framework. Our choice of the Coulomb gauge ensures this—the nuclear positions only appear in the electron-nucleus interaction terms, and the derivation of this potential from relativistic field theory shows us that it is indeed the quantities of normal 3-space that appear here. Thus, any potential surface that we might calculate on the basis of the Born-Oppenheimer-separated electronic molecular Dirac equation is indeed spanned by the variations of molecular structural parameters in the usual meaning. 

Computational chemistry is now a standard part of chemical research. One major application is in pharmaceutical chemistry, where the likely pharmacological activity of a molecule can be assessed computationally from its shape and electron density distribution before expensive clinical trials are started. Commercial software is now widely available for calculating the electronic structures of molecules and displaying the results graphically. All such calculations work within the Born-Oppenheimer approximation and express the molecular orbitals as linear combinations of atomic orbitals. 

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