Wavefunctions Born-Oppenheimer approximation

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... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

In the Born-Oppenheimer approximation, the total wavefunction of a molecule is approximated as a product of parts which describe the translational motion, the rotational motion, the vibrational motion, the electronic motion, etc. According to the (approximate) Franck-Condon... 

The basic problem is to solve the time-independent electronic Schrodinger equation. Since the mass of the electrons is so small compared to that of the nuclei, the dynamics of nuclei and electrons can normally be decoupled, and so in the Born-Oppenheimer approximation the many-electron wavefunction P and corresponding energy may be obtained by solving the time-independent Schrodinger equation in which the nuclear positions are fixed. We thus solve... 

The methods described above are all based on the Born-Oppenheimer approximation. Therefore, they can be used to calculate polarizabilities of diatomic molecules for a given internuclear distance R. However, if one is interested in values of the polarizability tensors, and C", for a particular vibrational state /i )), one has to average the polarizability radial functions a(R) and C(R) with the vibrational wavefunction i.e., one has to... 

We restrict ourselves to the clamped-nucleus or Born-Oppenheimer approximation [30,31] because essentially all the work done to date on electron momentum densities has relied on it. Therefore we focus on purely electronic wavefunctions and the electron densities that they lead to. 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

The Born-Oppenheimer approximation may then be thought of as keeping the electronic eigenfunctions independent and not allowing them to mix under the nuclear coordinates. This may be seen by expanding the total molecular wavefunction using the adiabatic eigenfunctions as a basis... 

CORRECT WAVEFUNCTIONS FOR PERTURBATIONS (SPIN-ORBIT, EXTERNAL FIELD, RELATIVISTIC, ETC.) WITHIN BORN-OPPENHEIMER APPROXIMATION... 

Here p is the density of vibrational levels of states Sj and Sf at the energy of the electronic transition E. The overlap of the electronic wavefunctions 0i5 0f and of the vibrational wavefunctions (0i 0f) are factorized according to the Born-Oppenheimer approximation just as in the case of radiative transitions. The density of vibrational levels is greater for the lower (final) state Sf... 

The relative motion (slow degree of freedom) is analogous to nuclear motion in the Born-Oppenheimer approximation and the fast vibrational motion is analogous to electronic motion. The nuclear wavefunction ij> (p,q) can be written as a product... 

In the framework of the Born-Oppenheimer approximation, the motion of the nuclei in a molecule being in the zth electronic state characterized by the energy surface E R) and the wavefunction (x, R) is determined by the shape of the surface. After preparation of... 

Within the Born-Oppenheimer approximation the time-independent molecular wavefunctions for the various electronic states are written as... 

After having defined the partial dissociation wavefunctions l>(R,r E,n) as basis in the continuum, the derivation of the absorption rates and absorption cross sections proceeds in the same way as outlined for bound-bound transitions in Sections 2.1 and 2.2. In analogy to (2.9), the total time-dependent molecular wavefunction T(t) including electronic (q) and nuclear [Q = (R, r)] degrees of freedom is expanded within the Born-Oppenheimer approximation as... 

So and Si are the electronic wavefunctions in states 0 and 1, respectively. 0 is the initial nuclear wavefunction in the electronic ground state and the 4/i are the bound vibrational wavefunctions in electronic state 1. The corresponding energies are Eo and Ei, respectively. The vector q comprises all electronic coordinates. 4>o(t) and i(t) represent the corresponding nuclear wavepackets in state 0 and 1, respectively. Equations (16.1) and (16.2) are the analogues of (2.9). Within the Born-Oppenheimer approximation it is assumed that there is no nonadiabatic coupling between the two electronic manifolds. 

Within the Born-Oppenheimer approximation, the Schrodinger equation for a whole molecular system can be divided into two equations. The electronic Schrodinger equation needs to be solved separately for each different (fixed) set of positions for the nuclei making up the system and gives the electronic wavefunction and the electronic... 

The Born-Oppenheimer Approximation.—The total wavefunction for a molecule may be approximated by separation into two factors,... 

In both cases we can introduce a similar picture in terms of an effective Hamiltonian giving rise to an effective Schrodinger equation for the solvated solute. Introducing the standard Born-Oppenheimer approximation, the solute electronic wavefunction ) will satisfy the following equation ... 

So far, this discussion of selection rules has considered only the electronic component of the transition. For molecular species, vibrational and rotational structure is possible in the spectrum, although for complex molecules, especially in condensed phases where collisional line broadening is important, the rotational lines, and sometimes the vibrational bands, may be too close to be resolved. Where the structure exists, however, certain transitions may be allowed or forbidden by vibrational or rotational selection rules. Such rules once again use the Born-Oppenheimer approximation, and assume that the wavefunctions for the individual modes may be separated. Quite apart from the symmetry-related selection rules, there is one further very important factor that determines the intensity of individual vibrational bands in electronic transitions, and that is the geometries of the two electronic states concerned. Relative intensities of different vibrational components of an electronic transition are of importance in connection with both absorption and emission processes. The populations of the vibrational levels obviously affect the relative intensities. In addition, electronic transitions between given vibrational levels in upper and lower states have a specific probability, determined in part... 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

Figure 18a depicts the harmonic potentials for a model system with one normal coordinate Q (vibrational coordinate) in the electronic ground state 0 and the excited state II. It is assumed that the potentials are (nearly) equal, but are shifted by AQ with respect to each other, i.e. the vibrational frequencies of both states are also (nearly) equal. Generally, the wavefunctions of electronic-vibrational states depend on both, the coordinates of the electrons and nuclei, and thus the wavefunctions are very difficult to handle. However, when it is taken into account that the electronic motion is much faster than the vibrational motion, one can factorize the vibrational and the electronic part of the wavefunction. This leads to the Born-Oppenheimer approximation with the... 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

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 the BO representation, the nuclear kinetic energy matrix is not diagonal because of the nuclear coordinate dependence of the wavefunction. The off-diagonal elements of the nuclear kinetic energy are non-adiabatic couplings. In order to discuss the relationship between vibronic coupling and non-adiabatic coupling, we present the Born-Oppenheimer approximation. 

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