Eigenfunctions completeness

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... 

The spin in quantum mechanics was introduced because experiments indicated that individual particles are not completely identified in terms of their three spatial coordinates [87]. Here we encounter, to some extent, a similar situation A system of items (i.e., distributions of electrons) in a given point in configuration space is usually described in terms of its set of eigenfunctions. This description is incomplete because the existence of conical intersections causes the electronic manifold to be multivalued. For example, in case of two (isolated) conical intersections we may encounter at a given point m configuration space four different sets of eigenfunctions (see Section Vni). 

It should be mentioned that if two operators do not commute, they may still have some eigenfunctions in common, but they will not have a complete set of simultaneous eigenfunctions. For example, the and Lx components of the angular momentum operator do not commute however, a wavefunction with L=0 (i.e., an S-state) is an eigenfunction of both operators. 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

A. If the two operators act on different coordinates (or, more generally, on different sets of coordinates), then they obviously commute. Moreover, in this case, it is straightforward to find the complete set of eigenfunctions of both operators one simply forms a product of any eigenfunction (say fk) of R and any eigenfunction (say gn) of S. The function fk gn is an eigenfunction of both R and S ... 

In solving the eigenvalue problem for the energy operator //op, we have previously always introduced a complete basic set Wlt in which the eigenfunction has been expanded ... 

Since the operators P commute with one another we can choose a representation in which every basis vector is an eigenfunction of all the P s with eigenvalue It should be noted that the specification of the energy and momentum of a state vector does not uniquely characterize the state. The energy-momentum operators are merely four operators of a complete set of commuting observables. We shall denote by afi the other eigenvalues necessary to specify the state. Thus... 

It should be borne in mind that the bond eigenfunctions actually are obtained from the expressions given in this paper by substituting for s the complete eigenfunction i nM(r,0,ip), etc. It is not necessary that the r. part of the eigenfunctions be identical the assumption made in the above treatment is that they do not affect the evaluation of the coefficients in the bond eigenfunctions. 

The Structures of Simple Molecules.—The foregoing considerations throw some light on the structure of very simple molecules in the normal and lower excited states, but they do not permit such a complete and accurate discussion of these questions as for more complicated molecules, because of the difficulty of taking into consideration the effect of several unshared and sometimes unpaired electrons. Often the bond energy is not great enough to destroy s-p quantization, and the interaction between a bond and unshared electrons is more important than between a bond and other shared electrons because of the absence of the effect of concentration of the eigenfunctions. 

The metal carbonyls Ni(CO)4, Fe(CO)s, and Cr(CO)6 are observed to be diamagnetic. This follows from the theoretical discussion if it is assumed that an electron-pair bond is formed with each carbonyl for the nine eigenfunctions available (3d64s4p3) are completely filled by the n bonds and 2(9 — n) additional electrons attached to the metal atom (n = 4, 5, 6). The theory also explains the observed composition of these unusual sub-... 

Strictly speaking, the complete eigenfunction for the molecule should be made antisymmetric before the charge densities at the various positions are calculated. It is easily shown, however, that this further refinement in the treatment does not alter the results obtained. 

If /f form the complete set of eigenfunctions of Ho with eigenvalues that is,... 

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