Schrodinger equation spherical polar coordinates

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

Atoms are spherical objects, and what we do is to write the electronic Schrodinger equation in spherical polar coordinates, to mirror the symmetry of the problem. 

The solutions of the Schrodinger equation in these spherical polar coordinates are described in many books. They can be factorised and have the following form ... 

Because it is more convenient mathematically, the coordinate system is changed from Cartesian to spherical polar coordinates (see Fig. 12.15) before the Schrodinger equation is solved. In the system of spherical polar coordinates a given point in space, specified by values of the Cartesian coordinates x, y, and z, is described by specific values of r, 6, and < >. 

The kinetic-energy operator separates in spherical polar coordinates into radial and angular observables given by (3.63). The Schrodinger equation for a local, central potential is therefore... 

The Schrodinger equation can be written out just as in Section 4.5, except in spherical polar coordinates, and the potential energy can be written as in Equation 3.2. The resulting equation is impressively complicated, but it nonetheless can be solved as in Section 4.6. The solution must be continuous in all three coordinates, and the radial portion must satisfy the boundary condition 1/7 0 as... 

As you have seen, the Schrodinger equation may be written in spherical polar coordinates using the usual transformation. As a result, we can write a radial part and an angular part for the wavefunction ik. As an example, let s look at the wavefunction for the Is and 2pz orbitals in the hydrogen atom. These orbitals have the quantum numbers n = 1, l = 0, mg = 0 and n = 2, = 1, mg = 0, respectively. 

Here, r denotes the scalar distance of the electron measured from the centre of mass (assumed to be at rest) of the complete system in the infinite proton mass approximation this corresponds to the electron-proton distance. The resulting reduced Schrodinger equation for the relative electronic motion is found to be soluble in several different orthogonal coordinate systems. In particular, in spherical polar coordinates (r, 0, ) referred to the centre of mass, the natural unconfined ranges of these variables are... 

By contrast, the strong electric field problem has appeared (perhaps prematurely) to be well understood. This view is reinforced by the fact that the Schrodinger equation for an atom in a strong electric field, although nonseparable in spherical polar coordinates (n and i are not good quantum numbers) does turn out to be separable in parabolic cylinder coordinates, given by... 

Solving this equation will not concern us, although it is useful to note that it is advantageous to work in spherical polar coordinates (Figure 1.4). When we look at the results obtained from the Schrodinger wave equation, we talk in terms of the radial and angular parts of the wavefunction,... 

The general case of the rotator free to move in three dimensions is more complicated, but is treated according to similar principles. The Schrodinger equation is first expressed in spherical polar coordinates, r, 6, and . For the rotation of a rigid body about its centre of gravity, r is constant and is included in a term representing the moment of inertia, /. The conditions for physically admissible solutions lead to the result... 

The kinetic energy operator,however,is almost separable in spherical polar coordinates, and the actual method of solving the differential equation can be found in a number of textbooks. The bound solutions (negative total energy) are called orbitals and can be classified in terms of three quantum numbers, n, I and m, corresponding to the three spatial variables r, d and q>. The quantum numbers arise from the boundary conditions on the wave function, i.e. it must be periodic in the 0 and q> variables, and must decay to zero as r oo. Since the Schrodinger equation is not completely separable in spherical polar coordinates, there exist the restrictions n > /> m. The n quantum number describes the size of the orbital, the / quantum number describes the shape of the orbital, while the m quantum number describes the orientation of the orbital relative to a fixed coordinate system. The / quantum number translates into names for the orbitals ... 

This form of the equation is not easily applied to rotational motion because the Cartesian coordinates used do not reflect the centro-symmetric nature of the problem. It is better to express the Schrodinger equation in terms of the spherical polar coordinates r, 6 and 0, which are shown in Figure 5.7. Their mathematical relationship to x. y and z is given on the left of the diagram. In terms of these coordinates the Laplacian operator becomes ... 

Although the Schrodinger equation looks much more formidable in spherical polar coordinates than it did in Cartesian coordinates, it is easier to solve because the wavefunctions can often be written as the product of three functions, each one of which involves only one of the... 

Because r is a constant, differentiation with respect to r can be ignored in equation (5.22) and the Schrodinger equation in spherical polar coordinates simplifies to ... 

In spherical polar coordinates the Schrodinger equation can now be written as ... 

However, a solution can be found in spherical polar coordinates. This solution can be represented as a product of functions each dependent on one coordinate. These are the coordinates r,6,(p, where r is the length of the segment connecting the electron with the nucleus (the origin), 0 and (p are the polar and azimuthal coordinates, respectively. Because of this simplification of the potential, it is possible to carry out the separation of variables in the time-independent Schrodinger equation. This now can be written instead of (3.2) as... 

This fact may be arrived at in two ways. One way is to write down the Schrodinger equation for using spherical polar coordinates or elliptical coordinates. (coordinate systems.) Then one atten ts to separate coordinates and finds that the 4> coordinate is indeed separable from the others and yields the equation... 

From Equation (A9.4) we see that the electron potential energy will have the same value at any point a distance r from the nucleus, i.e. any point on the surface of a sphere of radius r. Equation (A9.4) also shows that, in Cartesian coordinates, r actually depends on all three components of the axis system, which makes the direct solution of the Schrodinger equation quite difficult. However, if we transform to the spherical polar coordinate system illustrated in Eigure A9.1, then the distance from the origin r becomes a single coordinate. In our problem, the potential energy of the electron is then a much simpler function than in the Cartesian case. 

The relative Schrodinger equation can he solved in spherical polar coordinates hy separation of variables, assuming that... 

The relative Schrodinger equation cannot be solved in Cartesian coordinates. We transform to spherical polar coordinates in order to have an expression for the potential energy that contains only one coordinate. Spherical polar coordinates are depicted in Figure 17.3. The expression for the Laplacian operator in spherical polar coordinates is found in Eq. (B-47) of Appendix B. The relative Schrodinger equation is now... 

The time-independent Schrodinger equation for a hydrogen atom was separated into a one-particle Schrodinger equation for the motion of the center of mass of the two particles and a one-particle Schrodinger equation for the motion of the electron relative to the nucleus. The motion of the center of mass is the same as that of a free particle. The Schrodinger equation for the relative motion was solved by separation of variables in spherical polar coordinates, assuming the trial function... 

The electronic Schrodinger equation for the hJ ion can be solved by transforming to a coordinate system that is called confocal polar elliptical coordinates. One coordinate is f = (rA + rB)/rAB, the second coordinate isp = (rA — rB)/rAB, and the third coordinate is the angle , the same angle as in spherical polar coordinates. The solutions to the electronic Schrodinger equation are products of three factors ... 

We should not finish this discussion without mentioning the basis-set-fi-ee method of Becke. The grids used in this approach are the same as those described in Section 4.1. The grids were designed to accurately describe orbitals and densities in the neighborhood of each nucleus. A finite-difference approximation (in spherical polar coordinates) is used to solve Poisson s and Schrodinger s equations. The accuracy obtained with this basis-set-fi-ee approach for all-electron calculations is impressive, and, with the techniques described in Sections 3, 4.1 and 4,2, an 0(N) implementation is feasible. Delley s DMol program also uses a related approach. 

We shall be concerned mainly with the wavefunctions of electrons in atoms and in this case the predominant contribution to the potential comes from the Coulomb attraction of the nucleus. This potential is. spherically symmetric and therefore V(r) is a function of tlie radial coordinate r alone, This enables the Schrodinger equation to be separated into three differential equations which involve r, 0, and (j> separately. If we consider the motion of a single electron of mass m about a nucleus of mass M we can separate off the centre-of-mass motion and consider only the relative motion of the electron. In spherical polar coordinates equation C3.8) becomes (Problem 3.3)... 

---