Wave function polar form

In Figure 1.8 the real wave functions for the f, 2p and 3d orbitals are plotted in the form of polar diagrams, the construction of which may be illustrated by the simple case of the 2p orbital. The wave function in Equation (f.43) is independent of 4> and is simply proportional to cos 6. The polar diagram consists of points on a surface obtained by marking off, on lines drawn outwards from the nucleus in all directions, distances proportional to I cos 6 at a constant value of R2i(r). The resulting surface consists of two touching spheres. 

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

One may write the time-dependent wave function in the polar form, viz.,... 

This review has attempted to illustrate the relevance and the widespread utility of the CM model. Indeed, the author believes it is difficult to specify any area of structural or mechanistic chemistry where the CM approach is not applicable. The reason is not hard to find the CM model has its roots in the Schrodinger equation and as such its relevance to chemistry cannot be easily overstated. Even the fundamental chemical concept of a covalent bond derives from the CM approach. The covalent bond (e.g. in H2) owes its energy to the configuration mix HfiH <— H H. A wave-function for the hydrogen molecule based on just one spin-paired form does not lead to a stable bond. Both spin forms are necessary. Addition of ionic configurations improves the bond further and in the case of heteroatomic bonds generates polar covalent bonds. 

The wave functions for the hydrogen atom are known exactly. They are functions of the three spatial coordinates of the electron and take their most simple form when we choose these coordinates to be the polar coordinates shown in Figure 8.1 in relation to a set of Cartesian axes. The point at jt y, and z in Cartesian coordinates is fixed by r, the radial distance, OP, from the origin of the coordinate system (always considered positive) 0, the angle between the z axis and the line OP and the angle between the x axis and the projection of OP on the xy plane. 

The kinematics of the situation for the case of optical limit type (e,2e) experiments are illustrated in Fig. 2b (Fig. 1 b of Hamnett et al.23), which shows the direction of the ejected electron j as a function of the two polar angles x and y. The angle between j and the vector K is denoted by i//. Provided the forward scattering kinematics are such that K is small and we may approximate /(K) by /0m(0), then, as is well known, regardless of the detailed form of the continuum wave function, provided that k I is orthogonal to Pq 35... 

The total wave function of the dimer in the ground vibrational state can be represented as a product of wave functions for dimer rotation, internal rotation, and intermolecular exchange. The last of these depends on the polar angles 6l and d2 between the dimer axis and the molecular C3 axes and has the form... 

Exercise 6.1 Consider two radicals, R and X (not Na and Cl of course) combining to form a bond. Since the bond is polar covalent, its wave function will be dominated by the HL structure. Letting r and x represent the singly occupied orbitals of the two radicals, the unnormalized HL wave function is... 

As in the hydrodynamic model the Bohmian interpretation assumes a wave function in polar form,... 

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