Molecular system nuclei position

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

For a molecular system, 4 is a function of the positions f the electrons and thenu 1 within the molecule, which we will designate as r and R, respectively These h are a shorthand for the set of component vectors describing the position of each particle. We ll use subscripted versions of theig to denote the vector correspondin to a particular electron or nucleus r, andR/. Note that electrons are treated individually, while each nucleus is treated as an aggregate the component nucleon.s are not treated individually. 

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. 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

EFG Electric Field Gradient. The rapidity with which the electric field generated by a given molecular system is changing, usually evaluated at the position of a nucleus. 

The geometrical and electronic structure for molecular systems in general will depend on the balance between the different terms in the Hamiltonian i.e. electron-nucleus, electron-electron and nucleus-nucleus interaction including the valence as well as the core electrons of the constituent atoms. The full Hamiltonian for the molecular system is normally separated into a Hamiltonian Hn for the nuclei and another one Hgi for the electrons with fixed positions for the nuclei according to Born Oppenheimer approximation [31]. 

Instead of the usual Cartesians, we can also apply a complete and non-redundant set of the so-called internal coordinates. In order to understand their application in vibrational calculations, let us consider a molecular system consisting of N nuclei let the Cartesian displacement vectors of the nuclei bedi,d2.---./ A around their equilibrium geometry in the usual three-dimensional Euclidean space E3. (The expression for the n-th Cartesian displacement vector is dn = Pn — P% where p is the instantaneous position vector of the n-th nucleus, and /0° is the position vector of the same nucleus at equilibrium. Hereafter, these position vectors correspond to an arbitrary origin. Note that for simplicity, we omit the explicit use of the atomic masses, i.e., do not use mass-weighted Cartesians.) A single-point 5 of a hypothetical 3iV-dimensional space (5 Iftsw), defined as... 

In our QM systems, we have temporarily restricted ourselves to systems of one electron. If, in addition, our system were to have only one nucleus, then we would not need to guess wave functions, but instead we could solve Eq. (4.16) exactly. The eigenfunctions that are determined in that instance are the familiar hydrogenic atomic orbitals. Is, 2s, 2p, 3s, 3p, 3d, etc., whose properties and derivation are discussed in detail in standard texts on quantum mechanics. For the moment, we will not investigate the mathematical representation of these hydrogenic atomic orbitals in any detail, but we will simply posit that, as functions, they may be useful in the construction of more complicated molecular orbitals. In particular, just as in Eq. (4.10) we constructed a guess wave function as a linear combination of exact wave functions, so here we will construct a guess wave function as a linear combination of atomic wave functions (p, i.e.,... 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

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