Symmetry manifold

The Hartree—Fock problem in its simplest form (Hartree, 1927 Fock, 1930) consists in finding the best orbitals a) so that the configuration p) approximates as closely as possible the lowest-energy eigenstate of H in the symmetry manifold /j. This is done using the variation theorem. 

Consider a state /) in the space spanned by the eigenstates of a Hamiltonian H, which belong to the symmetry manifold j. If the variational energy... 

The theorem is the basis of the variational method of approximating the lowest eigenstate of a particular symmetry manifold. We choose a trial form of /), which is varied to minimise (f H f) with the constraint that (/I/) = 1. The form of /) that gives the minimum is the best approximation. 

Hartree—Fock calculations may be performed to find sets of orbitals describing the lowest-lying states of different symmetry manifolds of an atom. It is found that each different state has a closed-shell core whose orbitals are closely independent of the state. 

The multiconfiguration Hartree—Fock procedure is concerned with a particular symmetry manifold /j. It is therefore necessary to specify an eigenstate only by the principal quantum number n. The eigenstate ) is expanded in a set of Nr symmetry configurations r) that belong to the same manifold. That is they are eigenstates of parity and total angular momentum with quantum numbers /,y,m. 

If the states i ) and i) belong to different symmetry manifolds, characterised by the quantum numbers, j, then the Hamiltonian matrix element is zero. It is economical to consider the diagonalisation in a particular symmetry manifold and we will begin our discussion in this way. The basis states fk) are now symmetry configurations consisting of linear combinations of configurations which have the symmetry /j of the manifold. 

In one sense the hydrogen atom is a trivial case since the symmetry configurations are one-orbital determinants and in any case the exact eigenstates are known. However, we use it to illustrate the answer to a nontrivial question. How well can the lower-energy eigenstates of an atomic system be represented by an M/-dimensional square-integrable basis for each symmetry manifold We remember that a complete set of atomic states includes the ionisation continuum. 

The diagonalisation problem for the symmetry manifold (neglecting spin—orbit coupling) is... 

Between the 2s2s and 2soos states there is a sequence of resonances with Hartree—Fock configurations 2sns, n = 3,oo. They occur just below the n=2 threshold at 10.20 eV in Eq and condense to this energy. A similar sequence of resonances occurs just below each inelastic threshold. Similar sequences occur in the other symmetry manifolds with Hartree—Fock configurations consisting of different orbitals. 

The T-matrix element for a particular reaction may be represented perfectly generally by a linear combination of resonance terms (Bloch, 1957), most of which overlap considerably. The methods we have considered in chapter 7 may be put into this form. Most resonances in the expansion are artifacts of the expansion and have no individual physical manifestation, but some of the ones lowest in energy are isolated, at least from others in the same symmetry manifold. They appear as anomalies in the energy dependence of cross sections. 

In general the observed ion states can be grouped into symmetry manifolds, characterised by the quantum numbers /,/ We consider each symmetry manifold separately. The configuration-interaction basis for the target consists of symmetry configurations r), which are linear combinations, with symmetry /,y, of determinants formed from the set of orbitals a). 

X-ray absorption spectroscopy combining x-ray absorption near edge fine structure (XANES) and extended x-ray absorption fine structure (EXAFS) was used to extensively characterize Pt on Cabosll catalysts. XANES Is the result of electron transitions to bound states of the absorbing atom and thereby maps the symmetry - selected empty manifold of electron states. It Is sensitive to the electronic configuration of the absorbing atom. When the photoelectron has sufficient kinetic energy to be ejected from the atom It can be backscattered by neighboring atoms. The quantum Interference of the Initial... 

For conciseness, throughout this article it is understood that all the states and manifolds have well defined symmetry, so the corresponding labels (L, M/, 5, Ms-, parity) and projectors are omitted wherever this is possible without ambiguities. 

In the absence of damping (and in units where ( b = 1), the invariant manifolds bisect the angles between the coordinate axes. The presence of damping destroys this symmetry. As the damping constant increases, the unstable manifold rotates toward the Agu-axis, the stable manifold toward the A<7u-axis. In the limit of infinite damping the invariant manifolds coincide with... 

Nevertheless, despite the discouraging predictions of the early calculations, some attempts were made to apply the ligand field model to sandwich complexes. The first approach on these lines, due to Matsen (28), assumed a strong ligand field of D h symmetry, which was shown to split the d-orbital manifold into three levels characterised by the value of m. Unfortunately, the calculated energetic order for the metallocenes was found to be... 

FIGURE 7.2 Zero-field manifold for 5 = 2. The energy levels on the left hand are for axial symmetry (E = 0 and a = 0), that is, two non-Kramer s doublets and a singulet. The degeneracy of the doublets is lifted by addition of an E-term and subsequent addition of an a-term. 

Mj = Mjmax-this is the only case, which can, in principle, lead to a singledeterminant wave function. This situation may be encountered in some limited highly axial systems, such as Dy-0+ [37]. However, usual compounds do not have any site symmetry, which means that several projections of the /-manifold will be mixed by the low-symmetry components of the CF, leading to a multideterminantal character of the wave function. 

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