Mixing angle

In this method, one notes that real-valued solutions of the time-independent Hamiltonian of a 2 x 2 matrix form can be written in terms of an 0(<1), q), which is twice the mixing angle, such that the electronic component which is initially 1 is cos [0(4>, < )/2], while that which is initially 0 is sin [0(4>,<3r)/2]. For the second matrix form in Eq. (68) (in which, for simplicity f x) = 1), we get... 

Now, we discuss how the geometric phase is related to the mixing angle in this two-state model. We begin by writing Eq. (A.ll) as the gauge transformation... 

Comparing this equation with Eq. (75), it is seen that the mixing angle p is, up to an additive constant, identical to the relevant adiabatic-to-diabatic transformation—angle y ... 

Figure 10, The mixing angle P, for the H2D system, as a function of hyperspherical angle...

A value of 0 = 0° corresponds to a pure ground state, and 6 = 90° to a pure 3,2 ground state. Since the d orbital rotation matrix elements are different for the d and d -y orbitals, this will lead to a variation of the local g tensor of the Fe" site with the mixing angle d ... 

There are three ways of implementing the GP boundary condition. These are (1) to expand the wave function in terms of basis functions that themselves satisfy the GP boundary condition [16] (2) to use the vector-potential approach of Mead and Truhlar [6,64] and (3) to convert to an approximately diabatic representation [3, 52, 65, 66], where the effect of the GP is included exactly through the adiabatic-diabatic mixing angle. Of these, (1) is probably the most... 

Mixing angle, non-adiabatic coupling, two-state molecular system, H3 molecule, 104— 109... 

The linear combination of color gauge fields G and the mixing angle are defined by... 

Here we introduced the mixing-angle (t) defined by tan[2 (t)] = and Hamiltonian in diagonal form reads... 

Figure 5. Within a bend Darling-Dennison stack, Hi/ increases but AEj. increases more rapidly as vz increases, resulting in a gradually decreasing mixing angle, Hy/AEy.

Figure 6. The entire bend Darling-Dennison stack, which contains the ZOBS, pulls away from the other Darling-Dennison stacks as V2 increases. The different stacks are connected by the 3, 245 anharmonic resonance, Although the inter-stack Hy increase as V2 increases, the Hy increase more slowly than A -, thus the mixing angle decreases slowly as vz increases.

This angle defines the flavor mixing in processes. For small mixing angles the terms in the Lagrangian [Eq. (198)] assume the form... 

The other techniques can be divided into a few categories (see also Ref. [14c]). One of them goes back to the idea first coined by Mulliken [41], Hush [42] and Lichten [8] of using molecular properties to determine the diabatic basis and is actually based on the Werner-Meyer-Macfas-Riera formula [32,33] for the adiabatic-to-diabatic mixing angle in terms of electronic dipole moments. This... 

Such an angle a is called the mixing angle of the adiabatic-to-diabatic transformation (ADT) for the two-state problem. 

Equation (30) has the following general solution for the ADT mixing angle [37]... 

Consider the electronic dipole moment matrix M = (mH = ((1P"ilrl1fr/)))H. and project it on a given direction in 1R3. This gives the matrix m = (mkl)kl. Imposing a smoothness condition of this matrix within the two-state diabatic basis, one readily obtains the ADT mixing angle in terms of the electronic dipole moment [32, 33,37,68]... 

Let us further rotate the two-state adiabatic Hamiltonian by the mixing angle adipo e. This yields... 

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