Orbital angular momentum wave functions

The way we set up the Schrodinger equation makes it easiest to solve for the angular momentum wave functions in Table 3.1 first, but the Cartesian orbitals will be more useful when we get to molecules, where the orientation of the electron density along different Cartesian axes becomes important. 

Polarization functions are functions of a higher angular momentum than the occupied orbitals, such as adding d orbitals to carbon or / orbitals to iron. These orbitals help the wave function better span the function space. This results in little additional energy, but more accurate geometries and vibrational frequencies. 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

For typical lepton energies of a few MeV, the de Broglie wavelength is of order 100 times the nuclear radius and when orbital angular momentum is zero, one can use the allowed approximation for their wave functions... 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

With six electrons and six MOs removed from the active space, one is left with 6 electrons in 20 orbitals, a calculation that could be performed easily. Several calculations were thus done with different space and spin symmetry of the wave function. The resulting ground state was found to be a septet state with all six electrons having parallel spin, and the orbital angular momentum was high with A = 11. Spin-orbit calculations showed that the spin and orbital angular momenta combined to form an O = 8 state. The final label of the ground state is thus yOg. 

Quantitative theories for the chemical shift and nuclear spin-spin interaction were developed by Ramsey (113) soon after the experimental discoveries of the effects. Unfortunately the complete treatments of these effects involve rather detailed knowledge of the electronic structures of molecules and require evaluation of matrix elements of the orbital angular momentum between ground and excited electronic states. These matrix elements depend sensitively on the behavior of the wave function near... 

The molecular electronic wave functions ipe] are classified using the operators that commute with Hei. For diatomic (and linear polyatomic) molecules, the operator Lz for the component of the total electronic orbital angular momentum along the internuclear axis commutes with Hel (although L2 does not commute with tfel). The Lz eigenvalues are MLh,... 

Recall that linear molecules have Ah as the absolute value of the axial component of electronic orbital angular momentum the electronic wave functions are classified as 2,n,A,, ... according to whether A is 0,1,2,3,. Similarly, nuclear vibrational wave functions are classified as... 

In Section 9.3 we showed that for an orbital or state wave function having angular momentum quantum number / (or L) the character of the representation for which this forms a basis, under a symmetry operation that consists of rotation by an angle a, is given by... 

L is the quantum number specifying the total orbital angular momentum for the term, 5 the total spin angular momentum. Each of these momenta has components in any chosen direction, z say, which take on the integral values Lz, from L to -L, or S. from S to -S, respectively. There are 1L + 1 values of L, and 2S + 1 values of Sz, each with appropriate wave functions. Consequently, a term specified by L and S is (2L + + l)-fold degenerate. 

In Chapter 1, we introduced the concept of parity, the response of the wave function to an operation in which the signs of the spatial coordinates were reversed. As we indicated in our discussion of a decay, parity conservation forms an important selection rule for a decay. Emission of an a particle of orbital angular momentum / carries a parity change (— l/ so that 1+ —0+ or 2 0+ a decays are forbidden. In general, we find that parity is conserved in strong and electromagnetic interactions. 

A corollary of this is that for a system of particles the parity is even if the sum of the individual orbital angular momentum quantum numbers /, is even the parity is odd if Xlt is odd. Thus, the parity of each level depends on its wave function. An excited state of a nucleus need not have the same parity as the ground state. 

The symbol for the normal hydrogen molecule is and means the following (/) there is no net orbital angular momentum around the axis of the molecule (/ /) the two electron spins are paired (as they have to be if the Pauli Exclusion Principle is not to be violated) (iii) the sum of the f s is an even number (zero in this case) and leads to the subscript g, which means that the wave function does not change sign by inversion through the center of symmetry (otherwise the symbol u would be used). 

Because the orbital angular momentum of the positron-hydrogen system is zero for s-wave scattering, the total wave function is spherically symmetric and depends only on the three internal coordinates which specify the shape of the three-body system. The kinetic energy operator... 

The quantum number, m , originating from the 0(6) and Schrodinger wave equation, indicates how the orbital angular momentum is oriented relative to some fixed direction, particularly in a magnetic field. Thus, ml roughly characterizes the directions of maximum extension of the electron... 

The single-particle wave function for the free photoelectron may be expressed as an expansion in angular momentum partial waves characterized by an orbital angular momentum quantum number l and and associated quantum number X for the projection of l on the molecular frame (MF) z axis [22, 23, 63-66],... 

Covalent bonding depends on the presence of two atomic receptor sites. When the electron reaches one of these sites its behaviour, while in the vicinity of the atom, is described by an atomic wave function, such as the ip(ls), (l = 0), ground-state function of the H atom. Where two s-type wave functions serve to swap the valence electron the interaction is categorized as of a type. The participating wave functions could also be of p, (l = 1), or d, (/ = 2) character to form 7r or 6 bonds respectively. The quantum number l specifies the orbital angular momentum of the valence electron. A common assumption in bonding theory is that a valence electron with zero angular momentum can be accommodated in a p or d state if a suitable s-state is not available. The reverse situation is not allowed. 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

---