Degenerate states, zeroth-order

The three multiplet components of an excited triplet state are degenerate in zeroth order. We have therefore, in principle, the freedom of choosing these in their spherical or Cartesian forms. On the other hand, the spin-orbit split triplet levels will transform according to the irreps of the molecular point group. For a smooth variation of the wave function gradient with respect to the perturbation parameter X, we employ Cartesian triplet spin functions also in the unperturbed case and express them as ket vectors ... 

The unperturbed system is two hydrogen atoms. We have two zeroth-order functions consisting of products of hydrogen-atom wave functions, and these belong to a degenerate level. The correct ground-state zeroth-order function is the linear combination (13.101). 

Figure 1- Representatiori of degenerate states from nonrelativistic components, (a) Degenerate zeroth-order states at (b) Spin-orbit interaction splits 11 state, (c) With full spin-orbit...

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch.(7) In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Bom-Oppenheimer states. Henry and Kasha(8) and Jortner and co-workers(9-12) have pointed out that the Bom-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

The corresponding zeroth-order quantum-mechanical results are obtainable by regarding the vector of actions I as having components which, in units of % are integers. Thus, zero-order quantum-mechanical states that are compatible with the resonance condition (i.e., two separable states n and iT such that n - n = m) are degenerate,... 

The term symbols 1S and 3S are defined in the next section. For the states of the s2p configuration, we get zeroth-order wave functions similar to those of (1.256), except that there are 12 (=4x3) zeroth-order functions instead of 4 the factor 3 arises from the spatially degenerate 2px, 2py> and 2pz orbitals. 

A typical problem of interest at Los Alamos is the solution of the infrared multiple photon excitation dynamics of sulfur hexafluoride. This very problem has been quite popular in the literature in the past few years. (7) The solution of this problem is modeled by a molecular Hamiltonian which explicitly treats the asymmetric stretch ladder of the molecule coupled implicitly to the other molecular degrees of freedom. (See Fig. 12.) We consider the the first seven vibrational states of the mode of SF (6v ) the octahedral symmetry of the SF molecule makes these vibrational levels degenerate, and coupling between vibrational and rotational motion splits these degeneracies slightly. Furthermore, there is a rotational manifold of states associated with each vibrational level. Even to describe the zeroth-order level states of this molecule is itself a fairly complicated problem. Now if we were to include collisions in our model of multiple photon excitation of SF, e wou d have to solve a matrix Bloch equation with a minimum of 84 x 84 elements. Clearly such a problem is beyond our current abilities, so in fact we neglect collisional effects in order to stay with a Schrodinger picture of the excitation dynamics. 

The zeroth-order wavefunction of an excited state in an extended system cannot be described by a single determinant since the energies of the excited determinants form continuous bands. The states associated with the same band are degenerate since the energy difference between two adjacent states is an infinitesimal number. Thus, to calculate the energy spectrum of excited states, one needs to use degenerate MBPT, two-particle green function theory or a method like EOM-CC [5],... 

In order to correlate them, we can introduce a second (weaker) laser field of frequency coo — and the Rabi frequency IT < if which couples the degenerate transitions with dipole moments pn 7 and p22 v i as indicated in Fig. 19b. Treating the second field perturbatively, at zeroth order the coupling results in new doubly dressed states [63]... 

A and / are considered as the degenerate eigenfunctions of the absorbing molecule in the absence of a magnetic field. The functions that describe the states A and / and their corresponding zeroth-order energies are... 

The zeroth-order energy level is Iot- — 1 0 0 —) = 14) twofold degenerate— the corresponding vibronic states are loj-iT+lOO—) = 1) and Iot- iT—1 0 0 +) = 12). The zeroth-order energy is given by the expression (A.7). The hrst-order energy correction is... 

B The dynamic particle-hole self energy C The zeroth-order degenerate states D First order self energy... 

We will see later that these states, which we call /f-states, are in zeroth-order degenerate to the y-states and contribute to the dynamic part of the self energy. The fourth component of the primary states contributes in the... 

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