Projection operator spherical harmonic

Effective one-electron equations for the channel orbital functions can be obtained either by evaluating orbital functional derivatives of the variational functional S or more directly by projecting Eq. (8.3) onto the individual target states p. With appropriate normalizing factors, ((")/ TV) = if/ps. Equations for the radial channel functions fps(r) are obtained by projecting onto spherical harmonics and elementary spin functions. The matrix operator acting on channel orbitals is... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

The projection of this operator onto spherical harmonics and spin functions defines a radial operator mpq = gpq - ep8pq, where ep = E — Ep. 

Pfi denotes a projection operator on spinor spherical harmonics centred at the core k... 

It is noted that the projection operator is an element of a reducible tensorial set and that its action as a projection operator is limited to homogeneous pol5momials and spherical harmonics of the degree 1. 

Now, the projection operator O extracts (projects) the term out of fi, and analogously the unification operator yields the coefficient (Ap) to [normalized to 4 jr/(2 / + 1)] when operating on /i. The anal3dical form of the operator may be obtained by using either Eq. (15a) or Eq. (15b), giving the two forms which appear in Eq, (29). The latter form is particularly useful when the operators SUlf and are to be used on polynomials which are spherical harmonics... 

The projection operator P refers now to the spherical harmonics centered at the core A... 

Stone applied similar reasoning to the problem of a three-dimensional cluster. Here, the solutions of the corresponding free-particle problem for an electron-on-a-sphere are spherical harmonics. These functions should be familiar because they also describe the angular properties of atomic orbitals.Two quantum numbers, L and M, are associated with the spherical harmonics, Yim 0,total orbital angular momentum and its projection on the z-axis, respectively. It is more convenient to use the real linear combinations of Yim 9,(p)dinA (except when M = 0), and... 

An obvious candidate for the non-local part of the pseudopotential operator is the spherical harmonic projection operator ... 

ELECTRONIC STRUCTURE CALCULATIONS with the projection operator onto spinor spherical harmonics... 

The advantage of the above form is that it can be fit into conventional non-rela-tivistic codes since the two-component spinor projections have been eliminated and we obtain simple Im > projections involving ordinary spherical harmonics. However, the averaging method eliminates the spin-orbit operator. Fortunately, the spin-orbit operator itself can be expressed in terms of RECPs as shown by Hafner and Schwarz (1978,1979) and Ermler et al. (1981). This form is shown below. 

In this equation, Y is the spherical harmonic rotational wave-function for rotational quantum number N and projection Mn, is a phase factor that arises from the orbital motion of the electron about the internuclear axis, analogous to the term in the one-electron orbitals, and i/>o contains aU the other details of the electron distribution. The wavefunction in this problem does not conserve parity, however. Prove that the set of wavefunctions shown does have a fixed parity p = +1, by operating on this formula with I ... 

The generalized wavefunctions of the angular momentum operator are called spherical harmonics, and their symbol is Tim- They depend on two quantum numbers, and are eigenfunctions of the angular momentum and of its projection on the z axis. They obviously are three-dimensional functions, and their nodes are surfaces. In analogy with equation 3.21 ... 

The second term in equation 6.4 contains the angular momentum projection operator P[ based on spherical harmonics lm, I)... 

Here, the projection operator P[j is set up with spinor spherical harmonics ljm, I)... 

It follows that the spherical components of the electric dipole operator will transform like spherical harmonics of order unity under any arbitrary rotation of the coordinate system. A quantity which transforms under rotations like the spherical harmonic Y (0,( i) is said to be an irreducible tensor operator T of rank k and projection q where the projection quantum number can take any integer value from -k to +k. Since any arbitrary function of 0 and < can generally be expanded as a sum of spherical harmonics, it is usually possible to express any physical operator in terms of irreducible tensor operators. For instance, the electric quad-rupole moment operator defined by equation (4.45) can be shown to be a tensor of rank 2 (Problem 5.6). 

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