Born-Oppenheimer approximation separation

The Born-Oppenheimer approximation separates the molecular wave function into a product of electronic and nuclear wave functions ... 

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 ... 

The Born-Oppenheimer approximation separates the theoretical study of molecules into two parts, the electronic structure part, Eq. (2), and the nuclear motion part, Eq. (7). This has produced a separation among theoretical chemists themselves the electronic structure... 

The Born-Oppenheimer approximation separates the electronic and nuclear coordinates. For Hj, we figured out there are nine coordinates overall. The three coordinates of the electron are separated into the effective Hamiltonian. Once we have obtained the electronic energy as a function of R, we can add and solve the Schrodinger equation along the remaining six coordinates for motion of the nuclei ... 

The Born-Oppenheimer approximation separates out the terms that depend on the electron coordinates (Eqs. 5.8 and 5.9) ... 

The Born-Oppenheimer approximation separates the electronic and nuclear coordinates in the Schrodinger equation, allowing us to split the one big problem into two smaller problems. But the motions of the nuclei separate still further. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

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 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... 

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 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). 

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. 

The separation of the electronic and nuclear motions depends on the large difference between the mass of an electron and the mass of a nucleus. As the nuclei are much heavier, by a factor of at least 1800, they move much more slowly. Thus, to a good approximation the movement of the elections in a polyatomic molecule can be assumed to take place in the environment of the nuclei that are fixed in a particular configuration. This argument is the physical basis of the Born-Oppenheimer approximation. 

First, we shall consider the model where the intermolecular vibrations A—B and intramolecular vibrations of the proton in the molecules AHZ,+1 and BHZ2+1 may be described in the harmonic approximation.48 In this case, using the Born-Oppenheimer approximation to separate the motion of the proton from the motion of the other atoms for the symmetric transition, Eq. (16) may be... 

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... 

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. 

The purpose of most quantum chemical methods is to solve the time-independent Schrodinger equation. Given that the nuclei are much more heavier than the electrons, the nuclear and electronic motions can generally be treated separately (Born-Oppenheimer approximation). Within this approximation, one has to solve the electronic Schrodinger equation. Because of the presence of electron repulsion terms, this equation cannot be solved exactly for molecules with more than one electron. 

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 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. 

Representing the molecular potential energy as an analytic function of the nuclear coordinates in this fashion implicitly invokes the Born-Oppenheimer approximation in separating the very fast electronic motions from the much slower ones of the nuclei. 

All of the calculations have been performed at the experimental equilibrium distance R = 1.128 A, in order to enable a proper comparison with the EOM-CCSD reference. In so far as there are neither largely interacting excited states nor special reasons for expecting a breakdown of the Born Oppenheimer approximation, great changes in the MAE are not expected if one takes the (SC) SDCI ground state equilibrium value for Re which is Re = 1.140 A (very close to the CCSD value, as expected Cfr. table 1). We have performed a separate calculation of the whole set of VEE with the aug-cc-pVDZ basis set at the Rg distance, in any case. The results have not been included in table II for the sake of clarity, but the total MAE values where 2.34 eV for the MR-SDCI and 0.17 eV for (SC)2mR-SDCI. 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

Since a molecule consists of both nuclei and electrons, it may not be at all obvious why calculations can be performed on just the electrons. However, the huge differences in masses between nuclei and electrons justify the Born-Oppenheimer approximation, which allows the motions of the electrons to be separated from the motions of the nuclei. 

In the case of direct vibrational excitation, the vibrational transition probability is given by p, where are the intermediate and ground vibrational states, respectively, and is the vibrational transition moment. The electronic transition probability out of the intermediate state is < n < e ng e > n>, where are the excited and ground electronic states, respectively, and is the electronic dipole moment operator and vibrational state in the upper electronic state. Applying the Born-Oppenheimer approximation, where the nuclear electronic motion are separated, S can be presented as... 

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