Central Field Wavefunction

In a central field of force, the wavefunctions can always be written in terms of spherical harmonics. 

The reduction of the free-ion parameters has been ascribed to different mechanisms, where in general two types of models can be distinguished. On the one hand, one has the most often used wavefunction renormalisation or covalency models, which consider an expansion of the open-shell orbitals in the crystal (Jprgcnscn and Reisfeld, 1977). This expansion follows either from a covalent admixture with ligand orbitals (symmetry-restricted covalency mechanism) or from a modification of the effective nuclear charge Z, due to the penetration of the ligand electron clouds into the metal ion (central-field covalency mechanism). 

The Pauli principle allows each orbital Ygm(0, ) to be occupied at most by two electrons, one with spin up and the other with spin down. If we fill all the individual angular wavefunctions which are solutions of the independent electron central field equations for a given value of , by putting all 2(2 + 1) (the factor of 2 arises because there are two spin states) electrons into a given subshell, then the resulting charge shell, given by... 

One approximate method for obtaining Eq. 7.182 in a central field form wa.s introduced by Hartree and named the self-consistent field approach. This method regards each electron in a many-electron atom as moving in the temporarily fixed field of the remaining electrons. The system can now be described in terms of one-electron wavefunctions (or orbitals) j(rj). The non-Coulomb potential energy for the jth electron is then V y(ry) atid this contains the other electronic coordinates only as parameters. Vjirj) can be chosen to be spherically symmetric. The computational procedure is to solve the Schrodinger equation for every electron in its own central field and then to make the wavefunctions, so found, self-consistent with their potential fields. The complete wavefunction for the system is a product of the one-electron functions. 

In principle, any molecular wavefunction can be expanded in terms of a complete set of functions centred at any convenient point in space. This approach, which is referred to as the one-centre expansion method, the central-field approximation or the united-atom method, goes back to the earliest days of atomic and molecular physics. 

This central field Hamiltonian results in a Schrodinger equation which may be readily solved in polar coordinates with wavefunctions of the form as shown in eq. (3). 

Since the open-shell term in P does not possess spherical symmetry, the effective Hamiltonian will contain a non-spherical potential and as a result, even with initial orbitals of true central-field form (i.e. with spherical-harmonic angle dependence), the first cycle of an SCF iteration will destroy the symmetry properties of the orbitals—the solutions that give an improved energy will not be of pure s and p type but will be mixtures. This is a second example of a symmetry-breaking situation, akin to the spin polarization encountered in the UHF method. The resultant many-electron wavefunction will also lose the symmetry characteristic of a true spectroscopic state there will be a spatial polarization of the Is 2s core and the predicted ground state will no longer be of pure P type, just as in the UHF calculation there will be a spin polarization and the exact spin multiplicity of the many-electron state will be lost. Of course, the many-electron Hamiltonian does possess spherical symmetry (i.e. invariance under rotations around the nucleus), and the reason for the symmetry breaking lies at the level of the one-electron (i.e. IPM-type) model—the effective field in the 1-electron Hamiltonian is a fiction rather than a reality. 

A bound-state wavefunction for a single electron in a central field, corresponding either to an actual or hypothetical potential (for example the effective field used in an independent-particle model), is usually referred to as an atomic orbital. We denote the potential energy by V (r) (angle-independent), and the eigenvalue equation is thus... 

At this point, the HF-SCF approach makes an additional assumption beyond assuming that the wavefunction is a product of one-electron wavefunctions. It is assumed that the potential of an electron in an atom can be made into a function of r only. This is called the central-field approximation. The potential V(r,6i,f>i) is averaged over 6 and so that it is a function of ri only. 

Wavefunctions in L-S coupling. We recall that in the central field approximation, discussed in section 3.9, each electron in a many-electron atom is considered to move in a central potential independently of the others, at least in the first approximation. The i electron may then be described by the function (equation (3.71)) in terms of... 

In the central field approximation, solutions can be chosen such that the overall wave-function and energy of the system are sums of wavefunctions and energies of one-electron systems, as shown in Equation 1.5. 

The first two entries belong to the field of quantum chemistry. The calculation of the nuclear wavefunctions and their overlap is the central theme of molecular dynamics it is the subject of the following two sections. 

The term M is the central part of the charge distribution of the nucleus. If we have two isotopes with different neutron number, the charge distribution will be slightly changed, thus changing the electric field inside the nucleus. For electrons with a nonzero wavefunction at the nucleus, this difference will cause a shift (field shift) in the energy levels between the Isotopes. The field shift can in an approximate way be described as... 

In order to evidence the effects of the crystal field, the central ion must be described by a wavefunction built up with a sufficiently flexible basis set. We have selected the double-zeta set (DZ) of Slater type orbitals proposed by Clementi [3]. The wave-function for the isolated NO J ion with this basis set had been already published by us some years ago [4]. 

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