Diabatic bases

In principle, the time evolution of a particular linear superposition on the molecular base states will reflect a chemical process via the changes shown by the amplitudes. This represents a complete quantum mechanical representation of the chemical processes in Hilbert space. The problem is that the separability cannot be achieved in a complete and exact manner. One way to introduce a model that is able to keep as much as possible of the linear superposition principle is to use generalized electronic diabatic base functions. 

Once the electronic diabatic base set is obtained, the quantum state is a linear superposition where the PCB configuration enters parametrically as Ck amplitudes ... 

The GED approach is a general procedure based on the exact solutions to the n-electron system. Only one Hamiltonian is required at variance with the infinite Hamiltonian approach (defined on the parametric -space) characteristic of the BO scheme. All the base functions are expanded from a unique origin of the I-frame. The characteristics of the n-electrons diabatic base functions are independent from the positions taken by the sources of the external potential. 

If one had a complete basis set then, for the stationary geometry, the corresponding electronic waveftmctions obtained by actual calculation would be models for the diabatic base functions. The latter statement has to be understood in the sense that the set of nodal planes so obtained must be kept fixed. This was the procedure used to extract diabatic base states for cis and trans ethylene. 

Within the close-coupled formulation given above, the Schrodinger equation is solved by expansion of the wavefimction in the asymptotic [case (e)] basis [9,20,33]. This basis can be designated a diabaUc basis, since it is independent of the interparticle separation. Other diabatic bases are possible, for example a case (a) basis. The advantage of the case (e) basis is that the Hamiltonian is diagonal asymptotically, so that the close-coupled equations become uncoupled as R-. ... 

In general equation (23b) is not satisfied so that rigorous diabatic bases do not exist. As a consequence, (R), determined from equation (22), is path dependent and the circulation of f (R), the line integral of f (R) along a closed loop, does not vanish. Equation (23b) is difficult to evaluate numerically. However, the circulation of f is comparatively easy to evaluate and provides a computationally attractive, and in view of equation (22) an intuitively appealing, alternative to equation (23b). This point is discussed further in Section 6.5 and is addressed numerically in Section 8. 

The derivation of the D matrix for a given contour is based on first deriving the adiabatic-to-diabatic transformation matrix, A, as a function of s and then obtaining its value at the end of the arbitrary closed contours (when s becomes io). Since A is a real unitary matrix it can be expressed in terms of cosine and sine functions of given angles. First, we shall consider briefly the two special cases with M = 2 and 3. 

We intend to show that an adiabatic-to-diabatic transformation matrix based on the non-adiabatic coupling matrix can be used not only for reaching the diabatic fi amework but also as a mean to determine the minimum size of a sub-Hilbert space, namely, the minimal M value that still guarantees a valid diabatization. 

The obvious way to form a similarity between the Wigner rotation matrix and the adiabatic-to-diabatic transformation mabix defined in Eqs. (28) is to consider the (unbreakable) multidegeneracy case that is based, just like Wigner rotation matrix, on a single axis of rotation. For this sake, we consider the particular set of T matrices as defined in Eq. (51) and derive the relevant adiabatic-to-diabatic transfonnation matrices. In what follows, the degree of similarity between the two types of matrices will be presented for three special cases, namely, the two-state case which in Wigner s notation is the case, j =, the tri-state case (i.e.,7 = 1) and the tetra-state case (i.e.,7 = ). 

This reactivity pattern can be rationalized in terms of a diabatic model which is based upon the principle of spin re-coupling in valence (VB) bond theory [86]. In this analysis the total wavefunction is represented as a combination of two electronic configurations arising from the reactant (reaction coordinate. At the outset of the reaction, is lower in energy than [Pg.141]

For flow of some kind of surfactant solutions (Habon G solutions at concentration 530 and 1,060 ppm) in the tube of d = 1.07 mm in the range of Reynolds number based on solvent viscosity Re = 10-450, the increase of pressure drop in adiabatic and diabatic conditions was observed compared to that of pure water. 

Qu W, Mudawar I (2002) Prediction and measurement of incipient boiling heat flux in micro-channel heat sinks. Int J Heat Mass Transfer 45 3933-3945 Qu W, Mudawar I (2004) Measurement and correlation of critical heat flux in two-phase micro-channel heat sinks. Int J Heat Mass Transfer 47 2045-2059 Quiben JM, Thome JR (2007a) Flow pattern based two-phase pressure drop model for horizontal tubes. Part I. Diabatic and adiabatic experimental study. Int. J. Heat and Fluid Flow. 28(5) 1049-1059... 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

The calculation of the Td matrix, involving second derivatives of the electronie wave-functions, is more expensive and subject to numerical inaccuracy than that of Gd-A simple approximation for the third term in eq.(19) is based on a partial expansion of the identity operator in terms of the diabatic basis ... 

Song L, Gao J (2008) On the construction of diabatic and adiabatic potential energy surfaces based on ab initio valence bond theory. J Phys Chem A ASAP... 

---