Hamiltonian modes diagonalization

This Hamiltonian is diagonal in the local mode basis [Eq. (4.43)] with eigenvalues... 

The Hamiltonian is diagonal in Fourier space, except for the coupling between q and —q. (This coupling is shown below to result in the fact that H depends only on the amplitude h(q). ) Thus by inspection (/r(5)) = 0 and the mean-square fluctuation of a mode with wavevector k, is given by... 

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... 

Now, it may be of interest to look at the connection between the autocorrelation functions appearing in the standard and the adiabatic approaches. Clearly, it is the representation I of the adiabatic approach which is the most narrowing to that of the standard one [see Eqs. (43) and (17)] because both are involving the diagonalization of the matricial representation of Hamiltonians, within the product base built up from the bases of the quantum harmonic oscillators corresponding to the separate slow and fast modes. However, among the... 

The fact that both the local- and the normal-mode limits are contained within the algebraic approach allows one to study in a straightforward way the transition from one to the other. It is convenient to use, for this study, the local basis [Eq. (4.17)] and diagonalize the Hamiltonian for two identical bonds... 

Within van Roosmalen s scheme, it is not possible to construct simple diagonal Hamiltonians with the degeneracies required by bent normal-mode molecules. These molecules must therefore be dealt with by numerically diagonalizing the Hamiltonian matrix as discussed in the following sections. 

The local mode Hamiltonian (5.16) includes only the operator C[2 that is, interactions of the Casimir type between bonds 1 and 2. One may wish, in some cases, to include also interactions of this type between bonds 1 and 3, C13, and 2 and 3, C23. These can be included by diagonalizing the secular matrix obtained by evaluating the matrix elements of C13 and C23 in the basis (5.4). These matrix elements are given by (5.15a) and (5.15b). 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

There is a one to one correspondence between the imperturbed fi equencies CO, C0j j = 1,. .., N,. .. appearing in the Hamiltonian equivalent of the GLE (Eq. 3) and the normal mode frequencies. The diagonalization of the potential has been carried out exphcitly in Refs. 88,90,91. One finds that the imstable mode frequency A is the positive solution of the Kramers-Grote Hynes (KGH) equation (7). This identifies the solution of the KGH equation as a physical barrier fi-equency. 

A one-level system e) that can exchange its population with the bath states [/) represents the case of autoionization or photoionization. However, the above Hamiltonian describes also a qubit, which can undergo transitions between the excited and ground states e) and g), respectively, due to its off-diagonal coupling to the bath. The bath may consist of quantum oscillators (modes) or two-level systems (spins) with different eigenfrequencies. Typical examples are spontaneous emission into photon or phonon continua. In the RWA, which is alleviated in Section 4.4, the present formalism applies to a relaxing qubit, under the substitutions... 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

A parameterization method of the Hamiltonian for two electronic states which couple via nuclear distortions (vibronic coupling), based on density functional theory (DFT) and Slaters transition state method, is presented and applied to the pseudo-Jahn-Teller coupling problem in molecules with an s2-lone pair. The diagonal and off-diagonal energies of the 2X2 Hamiltonian matrix have been calculated as a function of the symmetry breaking angular distortion modes and r (Td)] of molecules with the coordination number CN = 3... 

In order to remove the driven term in the effective Hamiltonian H of the slow mode given by Eq. (47), without affecting the diagonal effective Hamiltonian H ... 

In this new representation /// given by Eq. (C.l), the effective Hamiltonians of the slow mode (46) corresponding to the ground state 0 ) of the fast mode is unmodified. On the other hand, the slow mode effective Hamiltonian (47), related to the situation where the fast mode has jumped into its first excited state 1 ), becomes diagonal. The II and III representations of these Hamiltonians are given in Table VI and the details of the calculations are reported in Appendix C. 

The different diagonal Hamiltonians H. ree, HBend. and H°, dealing, respectively, with the high frequency mode, the bending mode, and the H-bond bridge may be assumed to be the non-Hermitean ones given in Table IX, in which the zero-point... 

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