Hamiltonian energy-dependent

Hamiltonian contains (fe2/2me r ) 32/3y2 whereas the potential energy part is independent of Y, the energies of the moleeular orbitals depend on the square of the m quantum number. Thus, pairs of orbitals with m= 1 are energetieally degenerate pairs with m= 2 are degenerate, and so on. The absolute value of m, whieh is what the energy depends on, is ealled the X quantum number. Moleeular orbitals with = 0 are ealled a orbitals those with = 1 are 7i orbitals and those with = 2 are 5 orbitals. 

In the Hamiltonian Eq. (3.39) the first term is the harmonic lattice energy given by Eq. (3.12). It depends only on A iU, i.e., the part of the order parameter that describes the lattice distortions. On the other hand, the electron Hamiltonian Hcl depends on A(.v), which includes the changes of the hopping amplitudes due to both the lattice distortion and the disorder. The free electron part of Hel is given by Eq. (3.10), to which we also add a term Hc 1-1-1 that describes the Coulomb interne-... 

The hybrid Hamiltonian, which depends on both spatial and titration coordinates, can be written in terms of van der Waals, Coulomb, and GB electrostatic energies,... 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

The model is oversimplified in the sense that we have not attempted to specify what effects are incorporated in u. We will, however, consider the main effects from the vacuum fluctuations as well as other possible perturbations needed to produce the degeneracy above as well as, if necessary, considering the weak energy dependence in the Hamiltonian referred to in Eq. (11). To see how the CPT theorem affects our formulation we note that our zero order problem is an irreducible representation of... 

The effective hamiltonian is energy dependent, complex and nonlocal. To get a formal solution for P T >, the Q-projected hamiltonian (QHQ = Hqq) must be diagonalized. This amounts, in chemistry, to finding out stationary states for the supermolecule that might be relevant for describing different mechanistic pathways. 

The first part of the Hamiltonian (16), Hc.o.m, describes the center-of-mass contribution, as in the quasi-one-dimensional cases, and contributes the eigenenergy of a two-dimensional isotropic harmonic oscillator to the total energy. The second part of the Hamiltonian, Hint, depends on the antisymmetric coordinates xa and ya, and represents the contribution to the total energy due to the internal degrees of freedom. 

In the case of a scattering resonance, bound-free correlation is modified by a transient bound state of fV+1 electrons. In a finite matrix representation, the projected (fV+l)-electron Hamiltonian H has positive energy eigenvalues, which define possible scattering resonances if they interact sufficiently weakly with the scattering continuum. In resonance theory [270], this transient discrete state is multiplied by an energy-dependent coefficient whose magnitude is determined by that of the channel orbital in the resonant channel. Thus the normalization of the channel orbital establishes the absolute amplitude of the transient discrete state, and arbitrary normalization of the channel orbital cannot lead to an inconsistency. 

As mentioned earlier, it is highly desirable to get rid of the energy dependence of the effective Hamiltonians describing the subsystems. In order to do so we reconsider the general derivation of an effective Hamiltonian and specify it for the R-system. 

One-electron transfers between the subsystems finally contribute to the effective Hamiltonian the following energy dependent term ... 

Raleigh-Schrodinger perturbation theory and get rid of the energy dependence of the Hamiltonian as desired. Then the expression eq. (1.243) takes the form ... 

Multiplying the resonance integral by the quadrupled transferable spin bond order PoL = CQ- (3.14) results in the resonance energy of the m-th bond which is the only nontrivial contribution to the molecular energy at this (FAFO) level of approximate treatment of the MINDO/3 Hamiltonian using the SLG trial wave function. Within this picture the hybridization tetrahedra interact and the interaction energy depends on separations between centers of the tetrahedra, their mutual orientation, with respect to the bond axis. 

It is to be noted that the Hamiltonian matrix when constructed as described above will not transform properly if the orbitals are first hybridized by the usual procedure. This means that the calculated energies depend on whether the orbitals are hybridized or not. This situation arises for the following reason. A hybridized set of basis orbitals, 17,... 

One can adopt one of two viewpoints either to form the Hamiltonian in another way so that it will transform in the same way as S or to consider that the Hamiltonian matrices (as formed by Equations 6 or 7) are good approximations to the true Hartree-Fock matrix for the unhybridized basis, < in this case, in the transformed basis it must be T HT. We have chosen the latter viewpoint in these calculations. (J. A. Pople (17) has proposed that the off-diagonal elements be constructed as Hy — V2( + Pv)Sij where the fis are empirical energies depending only on the atom and not on the state of hybridization.)... 

The coupling to the continuum is implicitly contained in the second term in Eq. (376). The decay of the population in the bound state subspace is due to the imaginary part of P E — PHPY P. The contribution from the real part of P E — PHP) P, the so-called level shift due to the coupling to the continuum, can be neglected, if the energy dependence of the coupling term QHP is weak. In this case the second term in Eq. (376) can be regarded as a purely anti-Hermitian operator, and the effective Hamiltonian reduces to... 

These equations show that if we ignore AH e) and diagonalize Hin the subspace 5 we are in effect diagonalizing the projected Hamiltonian P HP thus the complementary subspace 5 is treated implicitly through the energy-dependent operator AH s). A useful way of thinking about the relationship between Eqs. (2-2) and (2-3) follows from the observation that ifis completely determined by the requirement that it should reproduce the exact eigenvalues E ,... 

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