Electrons Born-Oppenheimer

Separation of the wave functions describing the motion of nuclei and the wave function describing the electrons (Born-Oppenheimer approximation). This approximation is based on the fact that the nuclear particles are much heavier than the ele ctrons, and therefore much slower than the latter. In such a situation, the electronic wave function can be found at fixed positions of the nuclei. 

A practical representation of the full wave function, or the constituent orbitals, involves basis functions. The electronic wave function (or electron density) is parameterized by linear basis function coefficients or nonlinear parameters, such as the position or widths of Gaussian basis function. The array c will denote the collection of all wave function parameters, both linear and nonlinear unless otherwise specified. The ground electronic Born-Oppenheimer surface, is given by... 

Under the condition that there is a nontrivial MAB effect, let us ask what is its physical nature in relation to the standard AB effect Let us first address the issue of locality. As already indicated, the standard AB effect, which may occur when a charged particle encircles a line of magnetic flux, is nonlocal as it fulfills the criteria (Nl) and (N2) the effect arises although the particle experiences no physical field and no exchange of physical quantity takes place along the particle s path. Could the same be said about the MAB effect One way to address this question is to note that since the electronic Born-Oppenheimer states are eigenstates of it may be tempting to... 

The wave equation for an electronic Born-Oppenheimer state n is written ... 

As ab initio MD for all valence electrons [27] is not feasible for very large systems, QM calculations of an embedded quantum subsystem axe required. Since reviews of the various approaches that rely on the Born-Oppenheimer approximation and that are now in use or in development, are available (see Field [87], Merz ]88], Aqvist and Warshel [89], and Bakowies and Thiel [90] and references therein), only some summarizing opinions will be given here. 

The proper quantumdynamical treatment of fast electronic transfer reactions and reactions involving electronically excited states is very complex, not only because the Born-Oppenheimer approximation brakes down but... 

Hagedorn, G. A. Electron energy level crossing in the time-dependent Born-Oppenheimer approximation. Theor. Chim. Acta 67 (1990) 163-190... 

In the Car-Parrinello method [6] (and see, e.g., [24, 25, 16, 4]), the adiabatic time-dependent Born-Oppenheimer model is approximated by a fictitious Newtonian dynamics in which the electrons, represented by a set of... 

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

In currently available software, the Hamiltonian above is nearly never used. The problem can be simplified by separating the nuclear and electron motions. This is called the Born-Oppenheimer approximation. The Hamiltonian for a molecule with stationary nuclei is... 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

The total energy in an Molecular Orbital calculation is the net result of electronic kinetic energies and the interactions between all electrons and atomic cores in the system. This is the potential energy for nuclear motion in the Born-Oppenheimer approximation (see page 32). 

The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the... 

Since nuclei are much heavier than electrons and move slower, the Born-Oppenheimer Approximation suggests that nuclei are stationary and thus that we can solve for the motion of electrons only. This leads to the concept of an electronic Hamiltonian, describing the motion of electrons in the potential of fixed nuclei. 

Within the Born-Oppenheimer approximation discussed earlier, you can solve an electronic Schrodinger equation... 

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

The Born-Oppenheimer approximation allows the two parts of the problem to be solved independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term for the nuclei ... 

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 equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Evaluating the energy e for different values of R gives the effective potential for the nuclei in the presence of the electron. This function is called the Born-Oppenheimer potential surface or just the potential surface. In order to evaluate e(R) we have to determine HAA, HAB, and SAB. These quantities, which can be evaluated using elliptical coordinates, are given by... 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

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