Eigenfunctions characterized

In case of three conical intersections, we have as many as eight different sets of eigenfunctions, and so on. Thus we have to refer to an additional chai acterization of a given sub-sub-Hilbert space. This characterization is related to the number Nj of conical intersections and the associated possible number of sign flips due to different contours in the relevant region of configuration space, traced by the electronic manifold. 

The mathematical machinery needed to compute the rates of transitions among molecular states induced by such a time-dependent perturbation is contained in time-dependent perturbation theory (TDPT). The development of this theory proceeds as follows. One first assumes that one has in-hand all of the eigenfunctions k and eigenvalues Ek that characterize the Hamiltonian H of the molecule in the absence of the external perturbation ... 

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... 

In general, Eq. (4.2) has many acceptable eigenfunctions 4 for a given molecule, each characterized by a different associated eigenvalue E. That is, there is a complete set (perhaps infinite) of 4, with eigenvalues ,. For ease of future manipulation, we may assume without loss of generality that these wave functions are orthonormal, i.e., for a one particle system where the wave function depends on only three coordinates. 

All electrons are characterized by a spin quantum number. The electron spin function is an eigenfunction of the operator and has only two eigenvalues, h/2 the spin eigenfunctions... 

However, in most cases Ls is only a small perturbation on the energy. The wave functions are primarily constructed from functions which are eigenfunctions of L and S, so that we can characterize a given state by giving its L and 5 quantum numbers. The value of L is designated by a letter symbol ... 

Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions, however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character. 

Thus, in the central field approximation the wave function of the stationary state of an electron in an atom will be the eigenfunction of the operators of total energy, angular and spin momenta squared and one of their projections. These operators will form the full set of commuting operators and the corresponding stationary state of an atomic electron will be characterized by total energy E, quantum numbers of orbital l and spin s momenta as well as by one of their projections. 

The reference state of A-electron theory becomes a reference vacuum state 4>) in the field theory. A complete orthonormal set of spin-indexed orbital functions fip(x) is defined by eigenfunctions of a one-electron Hamiltonian Ti, with eigenvalues ep. The reference vacuum state corresponds to the ground state of a noninteracting A-electron system determined by this Hamiltonian. N occupied orbital functions (el < pi) are characterized by fermion creation operators a such that a] ) =0. Here pt is the chemical potential or Fermi level. A complementary orthogonal set of unoccupied orbital functions are characterized by destruction operators aa such that aa < >) = 0 for ea > p and a > N. A fermion quantum field is defined in this orbital basis by... 

In the case of periodic boundary conditions the chain Hamiltonian commutes with the operator that displaces all electrons by one unit cell cyclically. Therefore, its eigenfunctions must be characterized by the hole quasi-... 

It is a peculiarity of the Hartree-Fock scheme that the properties of the approximate eigenfunction Ca does not always reflect the properties of the exact eigenfunction C. A well-known example is given by the symmetry dilemma 8, which says that if an eigenfunction C has a special symmetry property characterized by the projector O. so that... 

The matrices D(P) with elements D(P) K form a matrix representation of the group of permutations and this representation is characterized by the labels S, Ms which indicate the total spin (S) and its z-component (Ms)- From (19) it follows that, when an orthonormal set of spin eigenfunctions is available, the matrix elements D(P) K may be expressed as... 

The stationary solutions are eigenfunctions of the time-independent wave equation (7), characterized by constant Vq. For an atom in an s-state (or any V o-state) the wave function is real, which means that the electron is at rest. This result may seem surprising, because classically a dynamic equilibrium is advanced to explain why the potential does not cause the particle to fall... 

Resonances are not truly bound states, but they are interpreted as metastable states. Because of the boundary conditions of resonances, the problem is not Hermitian even if the Hamiltonian is (that is, for square integrable eigenfunctions). Resonances are characterized by complex eigenvalues (complex poles of the scattering amplitude)... 

As far as electron-electron interaction is neglected, the Hamiltonian // a tt electron in a CNT commutes with all the element of G making, according to the basic theory of group representation [22], the electronic eigenfunctions a set of basis functions for the Irreducible representations of G. In fact, the basis functions = , of an irreducible representation of dimension I are characterized by the property... 

Even though in similarity to free particle wavefunctions the Bloch wavefunctions are characterized by the wavevector k, and even though Eq. (4.80) is reminiscent of free particle behavior, the functions V nkfr) are not eigenfunctions of the momentum operator. Indeed for the Bloch function (Eqs (4.78) and (4.79)) we have... 

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