Hamiltonian operator diagonalization

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

In an electronic adiabatic representation, however, the electronic Hamiltonian becomes diagonal,i.e. ( a 77e C/3) = da,0Va, where the adiabatic Va potentials for initial (A,B,B ) and final (X) electronic states were described in Ref.[31]. The couplings between different electronic states arises from the matrix elements of the nuclear kinetic operator Tn, giving rise to the so-called non-adiabatic coupling matrix elements (NACME) and are due to the dependence of the electronic functions on the nuclear coordinates. The actual form of these matrix elements depends on the choice of the coordinates. 

The different sets of coefficients that cause to satisfy Eq. 1, and the energy that corresponds to each wave function can be obtained by computing the energies of the interactions between each pair of configurations in Eq. 5, due to the Hamiltonian operator, H. If these interaction energies are displayed as a matrix, the coefficients that result in the MC wave function in Eq. 5 satisfying Eq. 1 are those that diagonalize this Hamiltonian matrix. 

H is the Hamiltonian operator and the numbering of the CSFs is arbitrary, but for convenience we will take I l = I hf and then all singly excited determinants, all doubly excited, etc. Solving the secular equation is equivalent to diagonalizing H, and permits determination of the CI coefficients associated with each energy. While this is presented without derivation, the formalism is entirely analogous to that used to develop Eq. (4.21). 

The seniority number (seniority for short), v, is a quantum number related to eigenvalues of the Racah seniority operator . The seniority matches the electron configuration ln in which the particular term first appears. We will see later that some matrix elements of the operators included in the Hamiltonian are diagonal in v so that their off-diagonal counterparts vanish. For some operators, however, there are crossing terms in v. 

In this chapter we introduce and derive the effective Hamiltonian for a diatomic molecule. The effective Hamiltonian operates only within the levels (rotational, spin and hyperfine) of a single vibrational level of the particular electronic state of interest. It is derived from the Ml Hamiltonian described in the previous chapters by absorbing the effects of off-diagonal matrix elements, which link the vibronic level of interest to other vibrational and electronic states, by a perturbation procedure. It has the same eigenvalues as the Ml Hamiltonian, at least to within some prescribed accuracy. 

It is conventional that the ligand field problem for systems with Na> d electrons requires the diagonalization of an effective Hamiltonian operator composed for the electronic kinetic energy T, and both one-electron ligand field terms, and two-electron Coulomb interactions ... 

The hyperspherical coordinate method is the subject of the present article. In this method, one coordinate, the hj-perradius, is treated by a propagator method. This leaves one coordinate less to treat by a bexsis set expansion than in the nri-ational approaches. Thus the eorresponding matrices are one dimension smaller. Here I will focxis on two aspects of the theory, viz. the application of boundary conditions and how the matrices can be diagonalized. A short derivation of a Hamiltonian operator for umbrella type motions is also inchided. I will end with some illustrations of calculations that we have performed and finally there will be some concluding remarks. 

Partial Diagonalization of the Three-Level Excitonic Interaction Hamiltonian Operator... 

In this section, we go beyond the perturbative treatment and calculate the eigenstates of large excitonic systems by diagonalizing the Hamiltonian operator on an efficient reduced basis set [164]. 

Enhancement of two-photon cross-sections by two-dimensional and three-dimensional arrangements of monomers has been demonstrated with fluorene V-shapes and dendrimeric structures. Such multidimensional structures lead to lower two-photon absorptions than Hnear oUgomers, but they have better one-photon transparencies. An accurate calculation of large exitonic systems is obtained by diagonalizing the Hamiltonian operator on a reduced basis set. 

In absence of external forces acting at the u-level, the set of fPui(u)Xi(R)Yi(p aoj) provides at least a subspace where the total Hamiltonian is diagonal since fp fu) is an eigenfunction of the momentum operator Pui. Now, we will describe physical and chemical processes with the help of this basis of orthogonal states. 

If these Born-Oppenheimer product wave functions are to approximate Hamiltonian eigenvectors, we have to minimize all off-diagonal matrix elements K L and u v). To this end, the electronic wave functions are chosen to be eigenvectors of a part of the Hamiltonian operator called the electronic Hamiltonian (adiabatic states) ... 

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